{
  "index": "1992-A-6",
  "type": "COMB",
  "tag": [
    "COMB",
    "GEO"
  ],
  "difficulty": "",
  "question": "What is the probability that the center of the sphere lies inside the\ntetrahedron whose vertices are at the four points? (It is understood that\neach point is independently chosen relative to a uniform distribution on\nthe sphere.)",
  "solution": "Solution 1. Set up a coordinate system so that the sphere is centered at the origin of \\( \\mathbb{R}^{3} \\). Identify points with vectors in \\( \\mathbb{R}^{3} \\).\n\nLet \\( P_{1}, P_{2}, P_{3}, P_{4} \\) be the four random points on the sphere. We may suppose that the choice of each \\( P_{i} \\) is made in two steps: first a random point \\( Q_{i} \\) is chosen, then a random sign \\( \\epsilon_{i} \\in\\{-1,1\\} \\) is chosen and we set \\( P_{i}=\\epsilon_{i} Q_{i} \\).\n\nThe probability that \\( Q_{3} \\) is in the linear subspace spanned by \\( Q_{1} \\) and \\( Q_{2} \\) is zero. Similar statements hold for any three of the \\( Q_{i} \\), so we may assume that every three of the \\( Q_{i} \\) are linearly independent. The probability that \\( Q_{4} \\) is in the plane through \\( Q_{1} \\), \\( Q_{2}, Q_{3} \\) is zero, so we may assume that \\( Q_{1} Q_{2} Q_{3} Q_{4} \\) is a nondegenerate tetrahedron \\( T_{Q} \\). We may also assume that for any choices of the \\( \\epsilon_{i} \\), the tetrahedron \\( T_{P} \\) with the \\( P_{i} \\) as vertices is nondegenerate.\n\nThe linear map \\( L_{Q}: \\mathbb{R}^{4} \\rightarrow \\mathbb{R}^{3} \\) sending \\( \\left(w_{1}, w_{2}, w_{3}, w_{4}\\right) \\) to \\( \\sum w_{i} Q_{i} \\) is surjective, so ker \\( L_{Q} \\) is 1-dimensional, generated by some \\( \\left(w_{1}, w_{2}, w_{3}, w_{4}\\right) \\). Since every three \\( Q_{i} \\) are linearly independent, every \\( w_{i} \\) is nonzero.\n\nWe claim that 0 lies in the interior of \\( T_{Q} \\) if and only if all the \\( w_{i} \\) have the same sign. If 0 lies in the interior of \\( T_{Q} \\), then 0 is a convex combination of the vertices in which each occurs with nonzero coefficient; i.e., \\( 0=\\sum_{i=1}^{4} r_{i} Q_{i} \\) for some \\( r_{1}, r_{2}, r_{3}, r_{4}>0 \\) with sum 1. Then \\( \\left(w_{1}, w_{2}, w_{3}, w_{4}\\right) \\) is a multiple of \\( \\left(r_{1}, r_{2}, r_{3}, r_{4}\\right) \\), so the \\( w_{i} \\) have the same sign. Conversely if the \\( w_{i} \\) have the same sign, then \\( w_{1}+w_{2}+w_{3}+w_{4} \\neq 0 \\) and \\( 0=\\sum_{i=1}^{4} r_{i} Q_{i} \\) where \\( \\left(r_{1}, r_{2}, r_{3}, r_{4}\\right)=\\frac{1}{w_{1}+w_{2}+w_{3}+w_{4}}\\left(w_{1}, w_{2}, w_{3}, w_{4}\\right) \\) is such that \\( r_{1}, r_{2}, r_{3}, r_{4}>0 \\) sum to 1 , so 0 lies in the interior of \\( T_{Q} \\).\n\nFix \\( Q_{1}, Q_{2}, Q_{3}, Q_{4} \\) and \\( \\left(w_{1}, w_{2}, w_{3}, w_{4}\\right) \\). Then \\( \\left(\\epsilon_{1} w_{1}, \\epsilon_{2} w_{2}, \\epsilon_{3} w_{3}, \\epsilon_{4} w_{4}\\right) \\) is a generator of ker \\( L_{P} \\), where \\( L_{P} \\) is defined using the \\( P_{i} \\) instead of the \\( Q_{i} \\). The numbers \\( \\epsilon_{1} w_{1} \\), \\( \\epsilon_{2} w_{2}, \\epsilon_{3} w_{3}, \\epsilon_{4} w_{4} \\) have the same sign (all plus or all minus) for exactly 2 of the \\( 2^{4}=16 \\) choices of \\( \\left(\\epsilon_{1}, \\epsilon_{2}, \\epsilon_{3}, \\epsilon_{4}\\right) \\). Thus, conditioned on a choice of the \\( Q_{i} \\), the probability that \\( T_{P} \\) contains the origin in its interior is \\( 2 / 16=1 / 8 \\). This is the same for any \\( \\left(Q_{1}, \\ldots, Q_{4}\\right) \\), so the overall probability that 0 lies inside \\( T_{P} \\) also is \\( 1 / 8 \\).\nSolution 2. Pick the first three points first; call them \\( A, B, C \\). Consider the spherical triangle \\( T \\) defined by those points. The center of the sphere lies in the convex hull of \\( A, B, C \\), and another point \\( P \\) if and only if it lies in the convex hull of \\( T \\) and \\( P \\). This happens if and only if \\( P \\) is antipodal to some point of \\( T \\). So the desired probability is the expected fraction of the sphere's surface area which is covered by \\( T \\).\n\nDenote the antipode to a point \\( P \\) by \\( P^{\\prime} \\). We consider the eight spherical triangles \\( A B C, A^{\\prime} B C, A^{\\prime} B^{\\prime} C, A B C^{\\prime}, A B^{\\prime} C^{\\prime}, A^{\\prime} B^{\\prime} C^{\\prime} \\). Denote these by \\( T_{0}, T_{1}, \\ldots, T_{7} \\); we regard \\( T_{i} \\) as a function of the random variables \\( A, B, C \\). There is an automorphism of our probability space defined by \\( (A, B, C) \\mapsto\\left(A^{\\prime}, B, C\\right) \\), so \\( T_{0} \\) and \\( T_{1} \\) have the same distribution. By choosing similar automorphisms, \\( T_{0} \\) and \\( T_{i} \\) have the same distribution for all \\( i \\). In particular, the expected fraction of the sphere covered by \\( T_{i} \\) is independent of \\( i \\). On the other hand, the triangles \\( T_{0}, \\ldots, T_{7} \\) cover the sphere (with overlap of measure zero), since they are the eight regions formed by the three great circles obtained by extending the sides of spherical triangle \\( A B C \\), so the probability we seek is \\( 1 / 8 \\).",
  "vars": [
    "P",
    "P_i",
    "Q_i",
    "\\\\epsilon_i",
    "w_i",
    "L_Q",
    "L_P",
    "r_i",
    "T_Q",
    "T_P",
    "T",
    "T_i",
    "A",
    "B",
    "C",
    "i",
    "R"
  ],
  "params": [],
  "sci_consts": [],
  "variants": {
    "descriptive_long": {
      "map": {
        "P": "pointp",
        "P_i": "pointpi",
        "Q_i": "pointqi",
        "\\epsilon_i": "signeps",
        "w_i": "weightwi",
        "L_Q": "linearq",
        "L_P": "linearp",
        "r_i": "ratioin",
        "T_Q": "tetraq",
        "T_P": "tetrap",
        "T": "triangle",
        "T_i": "triangli",
        "A": "pointa",
        "B": "pointb",
        "C": "pointc",
        "i": "indexi",
        "R": "realset"
      },
      "question": "What is the probability that the center of the sphere lies inside the\ntetrahedron whose vertices are at the four points? (It is understood that\neach point is independently chosen relative to a uniform distribution on\nthe sphere.)",
      "solution": "Solution 1. Set up a coordinate system so that the sphere is centered at the origin of \\( \\mathbb{realset}^{3} \\). Identify points with vectors in \\( \\mathbb{realset}^{3} \\).\n\nLet \\( pointp_{1}, pointp_{2}, pointp_{3}, pointp_{4} \\) be the four random points on the sphere. We may suppose that the choice of each \\( pointpi \\) is made in two steps: first a random point \\( pointqi \\) is chosen, then a random sign \\( signeps \\in\\{-1,1\\} \\) is chosen and we set \\( pointpi = signeps \\, pointqi \\).\n\nThe probability that \\( Q_{3} \\) is in the linear subspace spanned by \\( Q_{1} \\) and \\( Q_{2} \\) is zero. Similar statements hold for any three of the \\( pointqi \\), so we may assume that every three of the \\( pointqi \\) are linearly independent. The probability that \\( Q_{4} \\) is in the plane through \\( Q_{1}, Q_{2}, Q_{3} \\) is zero, so we may assume that \\( Q_{1} Q_{2} Q_{3} Q_{4} \\) is a nondegenerate tetrahedron \\( tetraq \\). We may also assume that for any choices of the \\( signeps \\), the tetrahedron \\( tetrap \\) with the \\( pointpi \\) as vertices is nondegenerate.\n\nThe linear map \\( linearq: \\mathbb{realset}^{4} \\rightarrow \\mathbb{realset}^{3} \\) sending \\( \\left(w_{1}, w_{2}, w_{3}, w_{4}\\right) \\) to \\( \\sum weightwi \\, pointqi \\) is surjective, so ker \\( linearq \\) is 1-dimensional, generated by some \\( \\left(w_{1}, w_{2}, w_{3}, w_{4}\\right) \\). Since every three \\( pointqi \\) are linearly independent, every \\( weightwi \\) is nonzero.\n\nWe claim that 0 lies in the interior of \\( tetraq \\) if and only if all the \\( weightwi \\) have the same sign. If 0 lies in the interior of \\( tetraq \\), then 0 is a convex combination of the vertices in which each occurs with nonzero coefficient; i.e., \\( 0=\\sum_{indexi=1}^{4} ratioin \\, pointqi \\) for some \\( r_{1}, r_{2}, r_{3}, r_{4}>0 \\) with sum 1. Then \\( \\left(w_{1}, w_{2}, w_{3}, w_{4}\\right) \\) is a multiple of \\( \\left(r_{1}, r_{2}, r_{3}, r_{4}\\right) \\), so the \\( weightwi \\) have the same sign. Conversely if the \\( weightwi \\) have the same sign, then \\( w_{1}+w_{2}+w_{3}+w_{4} \\neq 0 \\) and \\( 0=\\sum_{indexi=1}^{4} ratioin \\, pointqi \\) where \\( \\left(r_{1}, r_{2}, r_{3}, r_{4}\\right)=\\frac{1}{w_{1}+w_{2}+w_{3}+w_{4}}\\left(w_{1}, w_{2}, w_{3}, w_{4}\\right) \\) is such that \\( r_{1}, r_{2}, r_{3}, r_{4}>0 \\) sum to 1 , so 0 lies in the interior of \\( tetraq \\).\n\nFix \\( Q_{1}, Q_{2}, Q_{3}, Q_{4} \\) and \\( \\left(w_{1}, w_{2}, w_{3}, w_{4}\\right) \\). Then \\( \\left(\\epsilon_{1} w_{1}, \\epsilon_{2} w_{2}, \\epsilon_{3} w_{3}, \\epsilon_{4} w_{4}\\right) \\) is a generator of ker \\( linearp \\), where \\( linearp \\) is defined using the \\( pointpi \\) instead of the \\( pointqi \\). The numbers \\( \\epsilon_{1} w_{1}, \\epsilon_{2} w_{2}, \\epsilon_{3} w_{3}, \\epsilon_{4} w_{4} \\) have the same sign (all plus or all minus) for exactly 2 of the \\( 2^{4}=16 \\) choices of \\( \\left(\\epsilon_{1}, \\epsilon_{2}, \\epsilon_{3}, \\epsilon_{4}\\right) \\). Thus, conditioned on a choice of the \\( pointqi \\), the probability that \\( tetrap \\) contains the origin in its interior is \\( 2 / 16 = 1 / 8 \\). This is the same for any \\( \\left(Q_{1}, \\ldots, Q_{4}\\right) \\), so the overall probability that 0 lies inside \\( tetrap \\) also is \\( 1 / 8 \\).\n\nSolution 2. Pick the first three points first; call them \\( pointa, pointb, pointc \\). Consider the spherical triangle \\( triangle \\) defined by those points. The center of the sphere lies in the convex hull of \\( pointa, pointb, pointc \\), and another point \\( pointp \\) if and only if it lies in the convex hull of \\( triangle \\) and \\( pointp \\). This happens if and only if \\( pointp \\) is antipodal to some point of \\( triangle \\). So the desired probability is the expected fraction of the sphere's surface area which is covered by \\( triangle \\).\n\nDenote the antipode to a point \\( pointp \\) by \\( pointp^{\\prime} \\). We consider the eight spherical triangles \\( pointa pointb pointc, pointa^{\\prime} pointb pointc, pointa^{\\prime} pointb^{\\prime} pointc, pointa pointb pointc^{\\prime}, pointa pointb^{\\prime} pointc^{\\prime}, pointa^{\\prime} pointb^{\\prime} pointc^{\\prime} \\). Denote these by \\( triangle_{0}, triangle_{1}, \\ldots, triangle_{7} \\); we regard \\( triangli \\) as a function of the random variables \\( pointa, pointb, pointc \\). There is an automorphism of our probability space defined by \\( (pointa, pointb, pointc) \\mapsto\\left(pointa^{\\prime}, pointb, pointc\\right) \\), so \\( triangle_{0} \\) and \\( triangle_{1} \\) have the same distribution. By choosing similar automorphisms, \\( triangle_{0} \\) and \\( triangli \\) have the same distribution for all \\( indexi \\). In particular, the expected fraction of the sphere covered by \\( triangli \\) is independent of \\( indexi \\). On the other hand, the triangles \\( triangle_{0}, \\ldots, triangle_{7} \\) cover the sphere (with overlap of measure zero), since they are the eight regions formed by the three great circles obtained by extending the sides of spherical triangle \\( pointa pointb pointc \\), so the probability we seek is \\( 1 / 8 \\)."
    },
    "descriptive_long_confusing": {
      "map": {
        "P": "sandglass",
        "P_i": "hourglass",
        "Q_i": "lithograph",
        "\\\\epsilon_i": "watercress",
        "w_i": "roundhouse",
        "L_Q": "afterglow",
        "L_P": "columbine",
        "r_i": "blueberry",
        "T_Q": "orangewood",
        "T_P": "dragonfly",
        "T": "daffodil",
        "T_i": "undertone",
        "A": "snowflake",
        "B": "courseware",
        "C": "screwdriver",
        "i": "parchment",
        "R": "peppercorn"
      },
      "question": "What is the probability that the center of the sphere lies inside the\ntetrahedron whose vertices are at the four points? (It is understood that\neach point is independently chosen relative to a uniform distribution on\nthe sphere.)",
      "solution": "Solution 1. Set up a coordinate system so that the sphere is centered at the origin of \\( \\mathbb{R}^{3} \\). Identify points with vectors in \\( \\mathbb{R}^{3} \\).\n\nLet \\( sandglass_{1}, sandglass_{2}, sandglass_{3}, sandglass_{4} \\) be the four random points on the sphere. We may suppose that the choice of each \\( sandglass_{parchment} \\) is made in two steps: first a random point \\( lithograph_{parchment} \\) is chosen, then a random sign \\( watercress_{parchment} \\in\\{-1,1\\} \\) is chosen and we set \\( sandglass_{parchment}=watercress_{parchment} \\, lithograph_{parchment} \\).\n\nThe probability that \\( Q_{3} \\) is in the linear subspace spanned by \\( Q_{1} \\) and \\( Q_{2} \\) is zero. Similar statements hold for any three of the \\( lithograph_{parchment} \\), so we may assume that every three of the \\( lithograph_{parchment} \\) are linearly independent. The probability that \\( Q_{4} \\) is in the plane through \\( Q_{1} \\), \\( Q_{2}, Q_{3} \\) is zero, so we may assume that \\( Q_{1} Q_{2} Q_{3} Q_{4} \\) is a nondegenerate tetrahedron \\( orangewood \\). We may also assume that for any choices of the \\( watercress_{parchment} \\), the tetrahedron \\( dragonfly \\) with the \\( sandglass_{parchment} \\) as vertices is nondegenerate.\n\nThe linear map \\( afterglow: \\mathbb{R}^{4} \\rightarrow \\mathbb{R}^{3} \\) sending \\( \\left(w_{1}, w_{2}, w_{3}, w_{4}\\right) \\) to \\( \\sum roundhouse \\, lithograph_{parchment} \\) is surjective, so ker \\( afterglow \\) is 1-dimensional, generated by some \\( \\left(w_{1}, w_{2}, w_{3}, w_{4}\\right) \\). Since every three \\( lithograph_{parchment} \\) are linearly independent, every \\( roundhouse \\) is nonzero.\n\nWe claim that 0 lies in the interior of \\( orangewood \\) if and only if all the \\( roundhouse \\) have the same sign. If 0 lies in the interior of \\( orangewood \\), then 0 is a convex combination of the vertices in which each occurs with nonzero coefficient; i.e., \\( 0=\\sum_{parchment=1}^{4} blueberry \\, Q_{parchment} \\) for some \\( r_{1}, r_{2}, r_{3}, r_{4}>0 \\) with sum 1. Then \\( \\left(w_{1}, w_{2}, w_{3}, w_{4}\\right) \\) is a multiple of \\( \\left(r_{1}, r_{2}, r_{3}, r_{4}\\right) \\), so the \\( roundhouse \\) have the same sign. Conversely if the \\( roundhouse \\) have the same sign, then \\( w_{1}+w_{2}+w_{3}+w_{4} \\neq 0 \\) and \\( 0=\\sum_{parchment=1}^{4} blueberry \\, Q_{parchment} \\) where \\( \\left(r_{1}, r_{2}, r_{3}, r_{4}\\right)=\\frac{1}{w_{1}+w_{2}+w_{3}+w_{4}}\\left(w_{1}, w_{2}, w_{3}, w_{4}\\right) \\) is such that \\( r_{1}, r_{2}, r_{3}, r_{4}>0 \\) sum to 1, so 0 lies in the interior of \\( orangewood \\).\n\nFix \\( Q_{1}, Q_{2}, Q_{3}, Q_{4} \\) and \\( \\left(w_{1}, w_{2}, w_{3}, w_{4}\\right) \\). Then \\( \\left(\\epsilon_{1} w_{1}, \\epsilon_{2} w_{2}, \\epsilon_{3} w_{3}, \\epsilon_{4} w_{4}\\right) \\) is a generator of ker \\( columbine \\), where \\( columbine \\) is defined using the \\( sandglass_{parchment} \\) instead of the \\( lithograph_{parchment} \\). The numbers \\( \\epsilon_{1} w_{1}, \\epsilon_{2} w_{2}, \\epsilon_{3} w_{3}, \\epsilon_{4} w_{4} \\) have the same sign (all plus or all minus) for exactly 2 of the \\( 2^{4}=16 \\) choices of \\( \\left(\\epsilon_{1}, \\epsilon_{2}, \\epsilon_{3}, \\epsilon_{4}\\right) \\). Thus, conditioned on a choice of the \\( lithograph_{parchment} \\), the probability that \\( dragonfly \\) contains the origin in its interior is \\( 2 / 16=1 / 8 \\). This is the same for any \\( \\left(Q_{1}, \\ldots, Q_{4}\\right) \\), so the overall probability that 0 lies inside \\( dragonfly \\) also is \\( 1 / 8 \\).\n\nSolution 2. Pick the first three points first; call them \\( snowflake, courseware, screwdriver \\). Consider the spherical triangle \\( daffodil \\) defined by those points. The center of the sphere lies in the convex hull of \\( snowflake, courseware, screwdriver \\), and another point \\( sandglass \\) if and only if it lies in the convex hull of \\( daffodil \\) and \\( sandglass \\). This happens if and only if \\( sandglass \\) is antipodal to some point of \\( daffodil \\). So the desired probability is the expected fraction of the sphere's surface area which is covered by \\( daffodil \\).\n\nDenote the antipode to a point \\( sandglass \\) by \\( sandglass^{\\prime} \\). We consider the eight spherical triangles \\( snowflake \\, courseware \\, screwdriver, snowflake^{\\prime} \\, courseware \\, screwdriver, snowflake^{\\prime} \\, courseware^{\\prime} \\, screwdriver, snowflake \\, courseware \\, screwdriver^{\\prime}, snowflake \\, courseware^{\\prime} \\, screwdriver^{\\prime}, snowflake^{\\prime} \\, courseware^{\\prime} \\, screwdriver^{\\prime} \\). Denote these by \\( T_{0}, T_{1}, \\ldots, T_{7} \\); we regard \\( undertone \\) as a function of the random variables \\( snowflake, courseware, screwdriver \\). There is an automorphism of our probability space defined by \\( (snowflake, courseware, screwdriver) \\mapsto\\left(snowflake^{\\prime}, courseware, screwdriver\\right) \\), so \\( T_{0} \\) and \\( T_{1} \\) have the same distribution. By choosing similar automorphisms, \\( T_{0} \\) and \\( undertone \\) have the same distribution for all \\( parchment \\). In particular, the expected fraction of the sphere covered by \\( undertone \\) is independent of \\( parchment \\). On the other hand, the triangles \\( T_{0}, \\ldots, T_{7} \\) cover the sphere (with overlap of measure zero), since they are the eight regions formed by the three great circles obtained by extending the sides of spherical triangle \\( snowflake \\, courseware \\, screwdriver \\), so the probability we seek is \\( 1 / 8 \\)."
    },
    "descriptive_long_misleading": {
      "map": {
        "P": "infiniteplane",
        "P_i": "infiniteplane_{terminator}",
        "Q_i": "cubepoint_{terminator}",
        "\\epsilon_i": "largemagnitude_{terminator}",
        "w_i": "lightness_{terminator}",
        "L_Q": "nonlinearmap_{Q}",
        "L_P": "nonlinearmap_{infiniteplane}",
        "r_i": "negativecoef_{terminator}",
        "T_Q": "flatcircle_{Q}",
        "T_P": "flatcircle_{infiniteplane}",
        "T": "straightline",
        "T_i": "shortsegment_{terminator}",
        "A": "emptiness",
        "B": "nothingness",
        "C": "voidspace",
        "i": "terminator",
        "R": "nonrealfield"
      },
      "question": "What is the probability that the center of the sphere lies inside the\ntetrahedron whose vertices are at the four points? (It is understood that\neach point is independently chosen relative to a uniform distribution on\nthe sphere.)",
      "solution": "Solution 1. Set up a coordinate system so that the sphere is centered at the origin of \\( \\mathbb{R}^{3} \\). Identify points with vectors in \\( \\mathbb{R}^{3} \\).\n\nLet \\( infiniteplane_{1}, infiniteplane_{2}, infiniteplane_{3}, infiniteplane_{4} \\) be the four random points on the sphere. We may suppose that the choice of each \\( infiniteplane_{terminator} \\) is made in two steps: first a random point \\( cubepoint_{terminator} \\) is chosen, then a random sign \\( largemagnitude_{terminator} \\in\\{-1,1\\} \\) is chosen and we set \\( infiniteplane_{terminator}=largemagnitude_{terminator} cubepoint_{terminator} \\).\n\nThe probability that \\( cubepoint_{3} \\) is in the linear subspace spanned by \\( cubepoint_{1} \\) and \\( cubepoint_{2} \\) is zero. Similar statements hold for any three of the \\( cubepoint_{terminator} \\), so we may assume that every three of the \\( cubepoint_{terminator} \\) are linearly independent. The probability that \\( cubepoint_{4} \\) is in the plane through \\( cubepoint_{1} \\), \\( cubepoint_{2}, cubepoint_{3} \\) is zero, so we may assume that \\( cubepoint_{1} cubepoint_{2} cubepoint_{3} cubepoint_{4} \\) is a nondegenerate tetrahedron \\( flatcircle_{Q} \\). We may also assume that for any choices of the \\( largemagnitude_{terminator} \\), the tetrahedron \\( flatcircle_{infiniteplane} \\) with the \\( infiniteplane_{terminator} \\) as vertices is nondegenerate.\n\nThe linear map \\( nonlinearmap_{Q}: \\mathbb{R}^{4} \\rightarrow \\mathbb{R}^{3} \\) sending \\( \\left(lightness_{1}, lightness_{2}, lightness_{3}, lightness_{4}\\right) \\) to \\( \\sum lightness_{terminator} cubepoint_{terminator} \\) is surjective, so ker \\( nonlinearmap_{Q} \\) is 1-dimensional, generated by some \\( \\left(lightness_{1}, lightness_{2}, lightness_{3}, lightness_{4}\\right) \\). Since every three \\( cubepoint_{terminator} \\) are linearly independent, every \\( lightness_{terminator} \\) is nonzero.\n\nWe claim that 0 lies in the interior of \\( flatcircle_{Q} \\) if and only if all the \\( lightness_{terminator} \\) have the same sign. If 0 lies in the interior of \\( flatcircle_{Q} \\), then 0 is a convex combination of the vertices in which each occurs with nonzero coefficient; i.e., \\( 0=\\sum_{terminator=1}^{4} negativecoef_{terminator} cubepoint_{terminator} \\) for some \\( negativecoef_{1}, negativecoef_{2}, negativecoef_{3}, negativecoef_{4}>0 \\) with sum 1. Then \\( \\left(lightness_{1}, lightness_{2}, lightness_{3}, lightness_{4}\\right) \\) is a multiple of \\( \\left(negativecoef_{1}, negativecoef_{2}, negativecoef_{3}, negativecoef_{4}\\right) \\), so the \\( lightness_{terminator} \\) have the same sign. Conversely if the \\( lightness_{terminator} \\) have the same sign, then \\( lightness_{1}+lightness_{2}+lightness_{3}+lightness_{4} \\neq 0 \\) and \\( 0=\\sum_{terminator=1}^{4} negativecoef_{terminator} cubepoint_{terminator} \\) where \\( \\left(negativecoef_{1}, negativecoef_{2}, negativecoef_{3}, negativecoef_{4}\\right)=\\frac{1}{lightness_{1}+lightness_{2}+lightness_{3}+lightness_{4}}\\left(lightness_{1}, lightness_{2}, lightness_{3}, lightness_{4}\\right) \\) is such that \\( negativecoef_{1}, negativecoef_{2}, negativecoef_{3}, negativecoef_{4}>0 \\) sum to 1 , so 0 lies in the interior of \\( flatcircle_{Q} \\).\n\nFix \\( cubepoint_{1}, cubepoint_{2}, cubepoint_{3}, cubepoint_{4} \\) and \\( \\left(lightness_{1}, lightness_{2}, lightness_{3}, lightness_{4}\\right) \\). Then \\( \\left(largemagnitude_{1} lightness_{1}, largemagnitude_{2} lightness_{2}, largemagnitude_{3} lightness_{3}, largemagnitude_{4} lightness_{4}\\right) \\) is a generator of ker \\( nonlinearmap_{infiniteplane} \\), where \\( nonlinearmap_{infiniteplane} \\) is defined using the \\( infiniteplane_{terminator} \\) instead of the \\( cubepoint_{terminator} \\). The numbers \\( largemagnitude_{1} lightness_{1} \\), \\( largemagnitude_{2} lightness_{2}, largemagnitude_{3} lightness_{3}, largemagnitude_{4} lightness_{4} \\) have the same sign (all plus or all minus) for exactly 2 of the \\( 2^{4}=16 \\) choices of \\( \\left(largemagnitude_{1}, largemagnitude_{2}, largemagnitude_{3}, largemagnitude_{4}\\right) \\). Thus, conditioned on a choice of the \\( cubepoint_{terminator} \\), the probability that \\( flatcircle_{infiniteplane} \\) contains the origin in its interior is \\( 2 / 16=1 / 8 \\). This is the same for any \\( \\left(cubepoint_{1}, \\ldots, cubepoint_{4}\\right) \\), so the overall probability that 0 lies inside \\( flatcircle_{infiniteplane} \\) also is \\( 1 / 8 \\).\n\nSolution 2. Pick the first three points first; call them \\( emptiness, nothingness, voidspace \\). Consider the spherical triangle \\( straightline \\) defined by those points. The center of the sphere lies in the convex hull of \\( emptiness, nothingness, voidspace \\), and another point \\( infiniteplane \\) if and only if it lies in the convex hull of \\( straightline \\) and \\( infiniteplane \\). This happens if and only if \\( infiniteplane \\) is antipodal to some point of \\( straightline \\). So the desired probability is the expected fraction of the sphere's surface area which is covered by \\( straightline \\).\n\nDenote the antipode to a point \\( infiniteplane \\) by \\( infiniteplane^{\\prime} \\). We consider the eight spherical triangles \\( emptiness\\, nothingness\\, voidspace, emptiness^{\\prime}\\, nothingness\\, voidspace, emptiness^{\\prime}\\, nothingness^{\\prime}\\, voidspace, emptiness\\, nothingness\\, voidspace^{\\prime}, emptiness\\, nothingness^{\\prime}\\, voidspace^{\\prime}, emptiness^{\\prime}\\, nothingness^{\\prime}\\, voidspace^{\\prime} \\). Denote these by \\( shortsegment_{0}, shortsegment_{1}, \\ldots, shortsegment_{7} \\); we regard \\( shortsegment_{terminator} \\) as a function of the random variables \\( emptiness, nothingness, voidspace \\). There is an automorphism of our probability space defined by \\( (emptiness, nothingness, voidspace) \\mapsto\\left(emptiness^{\\prime}, nothingness, voidspace\\right) \\), so \\( shortsegment_{0} \\) and \\( shortsegment_{1} \\) have the same distribution. By choosing similar automorphisms, \\( shortsegment_{0} \\) and \\( shortsegment_{terminator} \\) have the same distribution for all \\( terminator \\). In particular, the expected fraction of the sphere covered by \\( shortsegment_{terminator} \\) is independent of \\( terminator \\). On the other hand, the triangles \\( shortsegment_{0}, \\ldots, shortsegment_{7} \\) cover the sphere (with overlap of measure zero), since they are the eight regions formed by the three great circles obtained by extending the sides of spherical triangle \\( emptiness\\, nothingness\\, voidspace \\), so the probability we seek is \\( 1 / 8 \\)."
    },
    "garbled_string": {
      "map": {
        "P": "flksreah",
        "P_i": "kqtmnvcz",
        "Q_i": "blxajydf",
        "\\\\epsilon_i": "rghpzeqw",
        "w_i": "tzdvplmh",
        "L_Q": "ycgxbroe",
        "L_P": "gfjqnsut",
        "r_i": "mcxdtqov",
        "T_Q": "nlswvury",
        "T_P": "kshjgepa",
        "T": "hrqvmodc",
        "T_i": "zljksmfe",
        "A": "pqldnxhe",
        "B": "yhrgvsow",
        "C": "zmbtfxqa",
        "i": "nvkosfle",
        "R": "schwpkju"
      },
      "question": "What is the probability that the center of the sphere lies inside the\ntetrahedron whose vertices are at the four points? (It is understood that\neach point is independently chosen relative to a uniform distribution on\nthe sphere.)",
      "solution": "Solution 1. Set up a coordinate system so that the sphere is centered at the origin of \\( \\mathbb{schwpkju}^{3} \\). Identify points with vectors in \\( \\mathbb{schwpkju}^{3} \\).\n\nLet \\( kqtmnvcz_{1}, kqtmnvcz_{2}, kqtmnvcz_{3}, kqtmnvcz_{4} \\) be the four random points on the sphere. We may suppose that the choice of each \\( kqtmnvcz \\) is made in two steps: first a random point \\( blxajydf \\) is chosen, then a random sign \\( rghpzeqw \\in\\{-1,1\\} \\) is chosen and we set \\( kqtmnvcz=rghpzeqw\\, blxajydf \\).\n\nThe probability that \\( blxajydf_{3} \\) is in the linear subspace spanned by \\( blxajydf_{1} \\) and \\( blxajydf_{2} \\) is zero. Similar statements hold for any three of the \\( blxajydf \\), so we may assume that every three of the \\( blxajydf \\) are linearly independent. The probability that \\( blxajydf_{4} \\) is in the plane through \\( blxajydf_{1}, blxajydf_{2}, blxajydf_{3} \\) is zero, so we may assume that \\( blxajydf_{1} blxajydf_{2} blxajydf_{3} blxajydf_{4} \\) is a nondegenerate tetrahedron \\( nlswvury \\). We may also assume that for any choices of the \\( rghpzeqw \\), the tetrahedron \\( kshjgepa \\) with the \\( kqtmnvcz \\) as vertices is nondegenerate.\n\nThe linear map \\( ycgxbroe: \\mathbb{schwpkju}^{4} \\rightarrow \\mathbb{schwpkju}^{3} \\) sending \\( \\left(tzdvplmh_{1}, tzdvplmh_{2}, tzdvplmh_{3}, tzdvplmh_{4}\\right) \\) to \\( \\sum tzdvplmh_{nvkosfle} blxajydf_{nvkosfle} \\) is surjective, so ker \\( ycgxbroe \\) is 1-dimensional, generated by some \\( \\left(tzdvplmh_{1}, tzdvplmh_{2}, tzdvplmh_{3}, tzdvplmh_{4}\\right) \\). Since every three \\( blxajydf \\) are linearly independent, every \\( tzdvplmh_{nvkosfle} \\) is nonzero.\n\nWe claim that 0 lies in the interior of \\( nlswvury \\) if and only if all the \\( tzdvplmh_{nvkosfle} \\) have the same sign. If 0 lies in the interior of \\( nlswvury \\), then 0 is a convex combination of the vertices in which each occurs with nonzero coefficient; i.e., \\( 0=\\sum_{nvkosfle=1}^{4} mcxdtqov_{nvkosfle} blxajydf_{nvkosfle} \\) for some \\( mcxdtqov_{1}, mcxdtqov_{2}, mcxdtqov_{3}, mcxdtqov_{4}>0 \\) with sum 1. Then \\( \\left(tzdvplmh_{1}, tzdvplmh_{2}, tzdvplmh_{3}, tzdvplmh_{4}\\right) \\) is a multiple of \\( \\left(mcxdtqov_{1}, mcxdtqov_{2}, mcxdtqov_{3}, mcxdtqov_{4}\\right) \\), so the \\( tzdvplmh_{nvkosfle} \\) have the same sign. Conversely if the \\( tzdvplmh_{nvkosfle} \\) have the same sign, then \\( tzdvplmh_{1}+tzdvplmh_{2}+tzdvplmh_{3}+tzdvplmh_{4} \\neq 0 \\) and \\( 0=\\sum_{nvkosfle=1}^{4} mcxdtqov_{nvkosfle} blxajydf_{nvkosfle} \\) where\n\\[\n\\left(mcxdtqov_{1}, mcxdtqov_{2}, mcxdtqov_{3}, mcxdtqov_{4}\\right)=\\frac{1}{tzdvplmh_{1}+tzdvplmh_{2}+tzdvplmh_{3}+tzdvplmh_{4}}\\left(tzdvplmh_{1}, tzdvplmh_{2}, tzdvplmh_{3}, tzdvplmh_{4}\\right)\n\\]\nis such that \\( mcxdtqov_{1}, mcxdtqov_{2}, mcxdtqov_{3}, mcxdtqov_{4}>0 \\) sum to 1, so 0 lies in the interior of \\( nlswvury \\).\n\nFix \\( blxajydf_{1}, blxajydf_{2}, blxajydf_{3}, blxajydf_{4} \\) and \\( \\left(tzdvplmh_{1}, tzdvplmh_{2}, tzdvplmh_{3}, tzdvplmh_{4}\\right) \\). Then \\( \\left(rghpzeqw_{1} tzdvplmh_{1}, rghpzeqw_{2} tzdvplmh_{2}, rghpzeqw_{3} tzdvplmh_{3}, rghpzeqw_{4} tzdvplmh_{4}\\right) \\) is a generator of ker \\( gfjqnsut \\), where \\( gfjqnsut \\) is defined using the \\( kqtmnvcz \\) instead of the \\( blxajydf \\). The numbers \\( rghpzeqw_{1} tzdvplmh_{1}, rghpzeqw_{2} tzdvplmh_{2}, rghpzeqw_{3} tzdvplmh_{3}, rghpzeqw_{4} tzdvplmh_{4} \\) have the same sign (all plus or all minus) for exactly 2 of the \\( 2^{4}=16 \\) choices of \\( \\left(rghpzeqw_{1}, rghpzeqw_{2}, rghpzeqw_{3}, rghpzeqw_{4}\\right) \\). Thus, conditioned on a choice of the \\( blxajydf \\), the probability that \\( kshjgepa \\) contains the origin in its interior is \\( 2 / 16=1 / 8 \\). This is the same for any \\( \\left(blxajydf_{1}, \\ldots, blxajydf_{4}\\right) \\), so the overall probability that 0 lies inside \\( kshjgepa \\) also is \\( 1 / 8 \\).\n\nSolution 2. Pick the first three points first; call them \\( pqldnxhe, yhrgvsow, zmbtfxqa \\). Consider the spherical triangle \\( hrqvmodc \\) defined by those points. The center of the sphere lies in the convex hull of \\( pqldnxhe, yhrgvsow, zmbtfxqa \\), and another point \\( flksreah \\) if and only if it lies in the convex hull of \\( hrqvmodc \\) and \\( flksreah \\). This happens if and only if \\( flksreah \\) is antipodal to some point of \\( hrqvmodc \\). So the desired probability is the expected fraction of the sphere's surface area which is covered by \\( hrqvmodc \\).\n\nDenote the antipode to a point \\( flksreah \\) by \\( flksreah^{\\prime} \\). We consider the eight spherical triangles \\( pqldnxhe\\, yhrgvsow\\, zmbtfxqa, pqldnxhe^{\\prime}\\, yhrgvsow\\, zmbtfxqa, pqldnxhe^{\\prime}\\, yhrgvsow^{\\prime}\\, zmbtfxqa, pqldnxhe\\, yhrgvsow\\, zmbtfxqa^{\\prime}, pqldnxhe\\, yhrgvsow^{\\prime}\\, zmbtfxqa^{\\prime}, pqldnxhe^{\\prime}\\, yhrgvsow^{\\prime}\\, zmbtfxqa^{\\prime} \\). Denote these by \\( hrqvmodc_{0}, hrqvmodc_{1}, \\ldots, hrqvmodc_{7} \\); we regard \\( zljksmfe \\) as a function of the random variables \\( pqldnxhe, yhrgvsow, zmbtfxqa \\). There is an automorphism of our probability space defined by \\( (pqldnxhe, yhrgvsow, zmbtfxqa) \\mapsto\\left(pqldnxhe^{\\prime}, yhrgvsow, zmbtfxqa\\right) \\), so \\( hrqvmodc_{0} \\) and \\( hrqvmodc_{1} \\) have the same distribution. By choosing similar automorphisms, \\( hrqvmodc_{0} \\) and \\( zljksmfe \\) have the same distribution for all \\( nvkosfle \\). In particular, the expected fraction of the sphere covered by \\( zljksmfe \\) is independent of \\( nvkosfle \\). On the other hand, the triangles \\( hrqvmodc_{0}, \\ldots, hrqvmodc_{7} \\) cover the sphere (with overlap of measure zero), since they are the eight regions formed by the three great circles obtained by extending the sides of spherical triangle \\( pqldnxhe\\, yhrgvsow\\, zmbtfxqa \\), so the probability we seek is \\( 1 / 8 \\)."
    },
    "kernel_variant": {
      "question": "Let $d\\ge 2$ and define \n\\[\nn:=d+2,\\qquad m:=d+1=n-1 .\n\\]\nChoose $n$ independent random points\n\\[\nX_{0},X_{1},\\dots ,X_{d+1}\\stackrel{\\text{i.i.d.}}{\\sim}\\mathcal N(0,I_{m})\n      \\qquad\\bigl(\\text{standard Gaussian vectors in }\\mathbb R^{m}\\bigr)\n\\]\nand denote by \n\\[\n\\Delta:=\\operatorname{conv}\\{X_{0},\\dots ,X_{d+1}\\}\\subset\\mathbb R^{m}\n\\]\nthe $(d+1)$–simplex they determine.  \nWith probability one the $X_{i}$ are affinely independent.\n\nDefine the event  \n\\[\n\\mathscr E_{d}:=\\bigl\\{0\\in\\operatorname{int}\\Delta\\bigr\\}\n              =\\Bigl\\{\n                 0=\\textstyle\\sum_{i=0}^{d+1}\\lambda_{i}X_{i},\\;\n                 \\lambda_{i}>0,\\;\n                 \\sum_{i=0}^{d+1}\\lambda_{i}=1\n               \\Bigr\\}.\n\\]\nWhenever $\\mathscr E_{d}$ occurs the barycentric vector  \n$\\Lambda:=(\\lambda_{0},\\dots ,\\lambda_{d+1})$ is unique and lives in the\nopen standard simplex\n\\[\n\\Delta_{d+1}:=\\Bigl\\{(x_{0},\\dots ,x_{d+1})\\in(0,1)^{d+2}\\colon\n                 \\sum_{i=0}^{d+1}x_{i}=1\\Bigr\\}.\n\\]\n\n(a)  Show that\n\\[\n\\mathbb P(\\mathscr E_{d})=2^{-(d+1)} .\n\\]\n\n(b)  {\\bf Conditional on $\\mathscr E_{d}$}, prove that $\\Lambda$ has the\n{\\em symmetric–radial} density\n\\[\nf_{\\Lambda}(\\lambda_{0},\\dots ,\\lambda_{d+1})\n   =C_{d}\n     \\Bigl(\\lambda_{0}^{2}+\\dots +\\lambda_{d+1}^{2}\\Bigr)^{-\\frac{d+2}{2}},\n     \\qquad (\\lambda_{0},\\dots ,\\lambda_{d+1})\\in\\Delta_{d+1},\n\\]\nwhere\n\\[\nC_{d}\n   =\\frac{\n        2^{\\frac{d}{2}}\\Gamma\\!\\bigl(\\tfrac{d+2}{2}\\bigr)\n        \\bigl(\\tfrac{2}{\\pi}\\bigr)^{\\frac{d+2}{2}}\n      }{\\displaystyle\n        \\int_{\\Delta_{d+1}}\n           \\bigl(x_{0}^{2}+\\dots +x_{d+1}^{2}\\bigr)^{-\\frac{d+2}{2}}\n          \\,\\mathrm dx}.\n\\]\n\n(c)  Put $M_{d}:=\\max_{0\\le i\\le d+1}\\lambda_{i}$.  \nShow that there are universal constants $0<c_{1}<c_{2}<\\infty$ such that\nfor every $d\\ge 2$\n\\[\n\\frac{c_{1}\\sqrt{\\log d}}{d}\n       \\;\\le\\;\n       \\mathbb E[M_{d}]\n       \\;\\le\\;\n       \\frac{c_{2}\\sqrt{\\log d}}{d},\n\\]\nso that $\\mathbb E[M_{d}]=\\Theta\\!\\bigl(\\sqrt{\\log d}\\,/\\,d\\bigr)$ as\n$d\\to\\infty$.\n\n\n--------------------------------------------------------------------",
      "solution": "Write again $n:=d+2$ and $m:=d+1=n-1$.\n\n--------------------------------------------------------------------\n(a)  Probability that the origin is contained  \n--------------------------------------------------------------------\n\nStep 1.  A fixed configuration.  \nFor a concrete realisation let  \n\\[\nG:=(Q_{0}\\;\\dots\\;Q_{n-1})\\in\\mathbb R^{m\\times n},\\qquad \nQ_{i}\\sim\\mathcal N(0,I_{m})\n\\]\nbe the matrix that has the {\\em unsigned} Gaussian vectors in its\ncolumns.  Because $m=n-1$, the rank of $G$ equals $m$ with probability\none, so $\\ker G$ is the one–dimensional subspace  \n\\[\n\\ker G=\\operatorname{span}\\{w\\},\\qquad \nw=(w_{0},\\dots ,w_{n-1})\\in\\mathbb R^{n}\\setminus\\{0\\}.\n\\]\nThe vector $w$ is unique up to a global sign.\n\nStep 2.  Re–introducing the random signs.  \nIndependently of $(Q_{i})$ choose  \n\\[\n\\varepsilon=(\\varepsilon_{0},\\dots ,\\varepsilon_{n-1})\n               \\in\\{-1,1\\}^{n}\\quad\\text{uniformly},\n\\qquad \nX_{i}:=\\varepsilon_{i}Q_{i}.\n\\]\nThe kernel of the signed matrix\n$(X_{0}\\;\\dots\\;X_{n-1})$ is spanned by the vector\n\\[\nw^{(\\varepsilon)}:=\n(\\varepsilon_{0}w_{0},\\dots ,\\varepsilon_{n-1}w_{n-1}).\n\\]\n\nStep 3.  When is the origin inside?  \nBy the standard characterisation of a simplex that surrounds the origin,\n\\[\n0\\in\\operatorname{int}\\operatorname{conv}(X_{0},\\dots ,X_{n-1})\n\\;\\Longleftrightarrow\\;\n\\text{all coordinates of }w^{(\\varepsilon)}\\text{ have the same sign}.\n\\]\nDenote $\\sigma_{i}:=\\operatorname{sgn}(w_{i})\\in\\{-1,1\\}$.  The\nrequirement $\\varepsilon_{i}w_{i}>0$ for every $i$ is equivalent to\n$\\varepsilon_{i}=\\sigma_{i}$ for every $i$; the alternative possibility\n$\\varepsilon_{i}w_{i}<0$ for every $i$ amounts to\n$\\varepsilon_{i}=-\\sigma_{i}$.  Consequently {\\em exactly two} sign\npatterns,\n\\[\n\\varepsilon=\\sigma:=(\\sigma_{0},\\dots ,\\sigma_{n-1})\n\\quad\\text{or}\\quad\n\\varepsilon=-\\sigma,\n\\]\nare favourable, while the remaining $2^{n}-2$ patterns are not.\n\nStep 4.  Conditional and unconditional probability.  \nSince $\\varepsilon$ is uniform on $\\{-1,1\\}^{n}$,\n\\[\n\\mathbb P\\bigl(\\mathscr E_{d}\\,\\bigl|\\,Q_{0},\\dots ,Q_{n-1}\\bigr)\n      =\\frac{2}{2^{n}}=2^{\\,1-n}.\n\\]\nThe right–hand side is deterministic, hence\n\\[\n\\boxed{\\;\n\\mathbb P(\\mathscr E_{d})\\,=\\,2^{\\,1-n}\\;=\\;2^{-(d+1)}\\;}\n\\]\nas required.  (The value coincides with the special case $m=n-1$ of\nWendel’s theorem.)\n\n--------------------------------------------------------------------\n(b)  Conditional density of the barycentric vector  \n--------------------------------------------------------------------\n\nStep 1.  A uniform positive kernel vector.  \nRotational invariance of the Gaussian law implies that $V$, the\n(normalised) generator of $\\ker G$, is uniform on the sphere\n$S^{\\,n-1}$.  Passing to the unique representative with positive\ncoordinates,\n\\[\nV^{+}\\in S^{\\,n-1}_{+}:=\n\\bigl\\{x\\in S^{\\,n-1}\\colon x_{i}>0\\bigr\\},\n\\]\nwe keep the uniform distribution on the {\\em spherical simplex}\n$S^{\\,n-1}_{+}$.\n\nStep 2.  Linking $V^{+}$ and $\\Lambda$.  \nOn the event $\\mathscr E_{d}$ the barycentric coordinates\nare given by scaling $V^{+}$:\n\\[\n\\Lambda\n  =\\frac{V^{+}}{\\sum_{i=0}^{d+1}V^{+}_{i}}\n  \\in\\Delta_{d+1}.\n\\tag{1}\n\\]\n\nStep 3.  Half–normal representation.  \nLet $H_{0},\\dots ,H_{d+1}\\stackrel{\\text{i.i.d.}}{\\sim}|N(0,1)|$ be\nindependent half–normal variables and put\n\\[\nW_{i}:=\\frac{H_{i}}\n            {\\bigl(\\sum_{j=0}^{d+1}H_{j}^{2}\\bigr)^{1/2}},\n            \\qquad 0\\le i\\le d+1.\n\\]\nThe vector $W=(W_{0},\\dots ,W_{d+1})$ is uniform on $S^{\\,n-1}_{+}$\n(Muirhead, \\emph{Multivariate Statistical Theory}, Section 2.1), hence\n$V^{+}\\stackrel{d}{=}W$.  Combining with (1) gives the stochastic\nrepresentation\n\\[\n\\boxed{\n\\Lambda\n   =\\Bigl(\\frac{H_{0}}{\\sum_{j}H_{j}},\\dots ,\n           \\frac{H_{d+1}}{\\sum_{j}H_{j}}\\Bigr)\n   \\quad\\Big|\\;\\mathscr E_{d}}\n\\tag{2}\n\\]\n\nStep 4.  Density calculation.  \nWrite\n\\[\nt:=\\sum_{j=0}^{d+1}H_{j},\\qquad\n\\lambda_{i}:=\\frac{H_{i}}{t},\\qquad\nA(\\lambda):=\\sum_{i=0}^{d+1}\\lambda_{i}^{2}.\n\\]\nThe bijection  \n$\\Phi:(0,\\infty)\\times\\Delta_{d+1}\\to(0,\\infty)^{n}$,\n$(t,\\lambda)\\mapsto (t\\lambda_{0},\\dots ,t\\lambda_{d+1})$\nhas Jacobian determinant $t^{\\,n-1}$.  The joint density of the\nhalf–normals is\n$p(h)=\\sqrt{2/\\pi}\\,\\exp(-h^{2}/2)\\mathbf 1_{(0,\\infty)}(h)$, hence\n\\[\ng(t,\\lambda)=\n  t^{\\,n-1}\\Bigl(\\tfrac{2}{\\pi}\\Bigr)^{n/2}\n  \\exp\\!\\bigl\\{-\\tfrac{t^{2}}{2}A(\\lambda)\\bigr\\},\n  \\qquad t>0,\\;\\lambda\\in\\Delta_{d+1}.\n\\]\nIntegration with the Gamma identity  \n$\\int_{0}^{\\infty}t^{\\,n-1}\\mathrm e^{-\\beta t^{2}}\\mathrm dt\n      =\\tfrac12\\beta^{-n/2}\\Gamma(n/2)$\ngives\n\\[\nf_{\\Lambda}(\\lambda)=\n  \\Bigl(\\tfrac{2}{\\pi}\\Bigr)^{n/2}\\,2^{\\,\\frac{n}{2}-1}\n  \\Gamma\\!\\bigl(\\tfrac{n}{2}\\bigr)\\,\n  A(\\lambda)^{-\\frac{n}{2}},\n  \\qquad \\lambda\\in\\Delta_{d+1}.\n\\]\nNormalising over $\\Delta_{d+1}$ yields the constant $C_{d}$ displayed in\nthe question, and the law is symmetric–radial.  \\hfill$\\square$\n\n--------------------------------------------------------------------\n(c)  Size of the largest barycentric coordinate  \n--------------------------------------------------------------------\n\nFrom (2) we have the exact identity  \n\\[\nM_{d}:=\\max_{0\\le i\\le d+1}\\lambda_{i}\n      =\\frac{\\displaystyle\\max_{0\\le i\\le n-1}H_{i}}\n             {\\displaystyle\\sum_{j=0}^{n-1}H_{j}}\n      =\\frac{M^{\\!*}_{n}}{S_{n}},\n\\qquad n=d+2,\n\\tag{3}\n\\]\nwhere\n\\[\nM^{\\!*}_{n}:=\\max_{0\\le j\\le n-1}H_{j},\n\\qquad\nS_{n}:=\\sum_{j=0}^{n-1}H_{j}.\n\\]\nDenote $\\mu:=\\mathbb E[H_{0}]=\\sqrt{2/\\pi}$ and note\n$\\operatorname{Var}(H_{0})=1-2/\\pi$.\n\nUpper bound.  \nBy a Chernoff inequality,\n$\\mathbb P(S_{n}\\le\\tfrac{\\mu}{2}n)\\le\\mathrm e^{-c_{0}n}$ for some\n$c_{0}>0$.  On the complementary event,\n$M_{d}\\le 2M^{\\!*}_{n}/(\\mu n)$.  Classical extreme–value theory for\nGaussians (e.g.\\ Leadbetter–Lindgren–Rootzén) gives\n$\\mathbb E[M^{\\!*}_{n}]\\le\\sqrt{2\\log n}+C_{1}$.  Splitting the\nexpectation,\n\\[\n\\mathbb E[M_{d}]\n   \\le\\frac{2}{\\mu n}\\bigl(\\sqrt{2\\log n}+C_{1}\\bigr)\n      +\\mathrm e^{-c_{0}n}\n   \\le\\frac{c_{2}\\sqrt{\\log d}}{d},\n\\]\nfor some universal $c_{2}$.\n\nLower bound.  \nPut  \n$b_{n}:=\\sqrt{2\\log n}-\\dfrac{\\log\\log n}{\\sqrt{2\\log n}}$.  Then\n$\\mathbb P(M^{\\!*}_{n}\\ge b_{n})\\ge\\tfrac34$ for all $n\\ge 3$.  A second\nChernoff bound gives\n$\\mathbb P(S_{n}\\le 2\\mu n)\\ge 1-\\mathrm e^{-c_{0}^{\\prime}n}\\ge\\tfrac34$,\nwhence, with probability at least $\\tfrac12$,\n\\[\nM_{d}\\ge\\frac{b_{n}}{2\\mu n}\n       \\ge c_{1}\\frac{\\sqrt{\\log n}}{n}\n       \\ge c_{1}\\frac{\\sqrt{\\log d}}{d},\n\\]\nfor a universal $c_{1}>0$.  Taking expectations yields the desired lower\nbound.\n\nCombining the two estimates we obtain\n\\[\n\\boxed{\\;\n\\frac{c_{1}\\sqrt{\\log d}}{d}\n       \\;\\le\\;\n       \\mathbb E[M_{d}]\n       \\;\\le\\;\n       \\frac{c_{2}\\sqrt{\\log d}}{d}}\n\\qquad(d\\to\\infty),\n\\]\nso $\\mathbb E[M_{d}]=\\Theta\\!\\bigl(\\sqrt{\\log d}\\,/\\,d\\bigr)$.\n\\hfill$\\square$\n\n\n--------------------------------------------------------------------",
      "metadata": {
        "replaced_from": "harder_variant",
        "replacement_date": "2025-07-14T19:09:31.724144",
        "was_fixed": false,
        "difficulty_analysis": "1.  Higher dimension and more variables – the problem is stated for arbitrary d ≥ 4 and involves the full (d+1)-tuple of barycentric coordinates, not just the yes/no event of containment.\n\n2.  Additional concepts – beyond probability of containment, parts (b) and (c) demand identification of an entire **probability distribution** (Dirichlet with parameter ½) and the evaluation of a non-trivial statistic (the expected maximum).\n\n3.  Deeper theoretical tools – the solution uses  \n  • linear-algebraic characterisation of convex containment,  \n  • orthogonal invariance and random-matrix reasoning,  \n  • relations between the uniform sphere, Gaussian vectors, chi-square laws, and Dirichlet distributions,  \n  • inclusion–exclusion on the probability simplex, incomplete Beta functions and hypergeometric identities.\n\n4.  Multiple interacting ideas – the chain “kernel of a random matrix → sign pattern → conditional Dirichlet law → order statistics of Dirichlet variables” forces the solver to blend geometry, probability, and special-function calculus.\n\nThese layers of technical sophistication go well beyond both the original problem (a single probability 1/8) and the current kernel variant (probability  that 0 lies in a 4-simplex), fulfilling the requirement of a **significantly harder** kernel variant."
      }
    },
    "original_kernel_variant": {
      "question": "Let $d\\ge 2$ and put\n\\[\nS^{d}:=\\bigl\\{x\\in\\mathbb R^{\\,d+1}\\colon\\lVert x\\rVert_{2}=1\\bigr\\},\n\\qquad n:=d+2 .\n\\]\nChoose $n$ independent random points\n\\[\nP_{0},P_{1},\\dots ,P_{d+1}\\stackrel{\\text{i.i.d.}}{\\sim}\\operatorname{Unif}(S^{d}),\n\\qquad \n\\Delta:=\\operatorname{conv}\\{P_{0},\\dots ,P_{d+1}\\}\\subset\\mathbb R^{\\,d+1}.\n\\]\nWith probability one the $P_{i}$ are affinely independent, hence $\\Delta$\nis a non–degenerate $(d+1)$–simplex.\nIntroduce the event\n\\[\n\\mathscr E_{d}:=\\bigl\\{0\\in\\operatorname{int}\\Delta\\bigr\\}\n  =\\Bigl\\{\n        0=\\textstyle\\sum_{i=0}^{d+1}\\lambda_{i}P_{i},\n        \\ \\lambda_{i}>0,\\\n        \\sum_{i=0}^{d+1}\\lambda_{i}=1\n    \\Bigr\\}.\n\\]\nWhenever $\\mathscr E_{d}$ occurs the vector\n$\\Lambda:=(\\lambda_{0},\\dots ,\\lambda_{d+1})$ is well–defined and lives\nin the open standard simplex\n\\[\n\\Delta_{d+1}:=\\Bigl\\{(x_{0},\\dots ,x_{d+1})\\in(0,1)^{d+2}\\colon\n          \\sum_{i=0}^{d+1}x_{i}=1\\Bigr\\}.\n\\]\n\n(a) Prove that\n\\[\n\\mathbb P(\\mathscr E_{d})=2^{-(d+1)} .\n\\]\n\n(b) {\\bf Conditional on $\\mathscr E_{d}$}, show that the barycentric\nvector $\\Lambda$ possesses the {\\em symmetric–radial} density\n\\[\nf_{\\Lambda}(\\lambda_{0},\\dots ,\\lambda_{d+1})\n      =C_{d}\\,\n        \\Bigl(\\lambda_{0}^{2}+\\dots+\\lambda_{d+1}^{2}\\Bigr)^{-\\frac{d+2}{2}},\n      \\qquad (\\lambda_{0},\\dots ,\\lambda_{d+1})\\in\\Delta_{d+1},\n\\]\nwhere\n\\[\nC_{d}\n  =\\frac{\n        2^{\\frac{d}{2}}\\Gamma\\!\\bigl(\\tfrac{d+2}{2}\\bigr)\n        \\bigl(\\tfrac{2}{\\pi}\\bigr)^{\\frac{d+2}{2}}\n      }{\\displaystyle\n        \\int_{\\Delta_{d+1}}\n             \\bigl(x_{0}^{2}+\\dots+x_{d+1}^{2}\\bigr)^{-\\frac{d+2}{2}}\n             \\,\\mathrm dx } .\n\\]\n\n(c) Put $M_{d}:=\\max_{0\\le i\\le d+1}\\lambda_{i}$.  Show that there exist\nuniversal constants $0<c_{1}<c_{2}<\\infty$ such that for every $d\\ge 2$\n\\[\n\\frac{c_{1}\\sqrt{\\log d}}{d}\n       \\;\\le\\;\n       \\mathbb E[M_{d}]\n       \\;\\le\\;\n       \\frac{c_{2}\\sqrt{\\log d}}{d},\n\\]\nso that $\\mathbb E[M_{d}]=\\Theta\\!\\bigl(\\sqrt{\\log d}\\,/\\,d\\bigr)$ as\n$d\\to\\infty$.\n\n\\bigskip",
      "solution": "Write $n:=d+2$ and $m:=d+1=n-1$ throughout.\n\n\\medskip\n{\\bf (a) Probability that the origin is contained.}\nWendel’s theorem states that for i.i.d.\\ random vectors\n$X_{1},\\dots ,X_{n}$ in $\\mathbb R^{m}$ whose common law is centrally\nsymmetric and has full support,\n\\[\n\\mathbb P\\!\\bigl(0\\notin\\operatorname{conv}\\{X_{1},\\dots ,X_{n}\\}\\bigr)\n     =2^{-(n-1)}\\sum_{k=0}^{m-1}\\binom{n-1}{k}.\n\\]\nBecause $\\operatorname{Unif}(S^{d})$ is centrally symmetric and\n$n=m+1$, we obtain\n\\[\n\\mathbb P\\!\\bigl(0\\notin\\Delta\\bigr)\n      =2^{-(d+1)}\\sum_{k=0}^{d}\\binom{d+1}{k}\n      =2^{-(d+1)}\\bigl(2^{d+1}-1\\bigr)=1-2^{-(d+1)},\n\\]\nand therefore $\\mathbb P(\\mathscr E_{d})=2^{-(d+1)}$.\n\n\\bigskip\n{\\bf (b) Density of $\\Lambda$ conditional on $\\mathscr E_{d}$.}\n\n\\emph{Step 1: A half–normal representation of the positive\nnull–vector.}\nLet $Z_{1},\\dots ,Z_{n}\\stackrel{\\text{i.i.d.}}{\\sim}N(0,1)$ and set\n\\[\nH_{i}:=|Z_{i}|,\\qquad\nW_{i}:=\\frac{H_{i}}{\\bigl(\\sum_{j=1}^{n}H_{j}^{2}\\bigr)^{1/2}},\n\\qquad 1\\le i\\le n.\n\\]\nThe vector $W:=(W_{1},\\dots ,W_{n})$ is uniform on the positive orthant\n\\[\nS^{\\,n-1}_{+}:=\\bigl\\{x\\in S^{\\,n-1}\\colon x_{i}>0\\bigr\\},\n\\]\nbecause $(Z_{1},\\dots ,Z_{n})$ is rotationally symmetric and the map\n$x\\mapsto |x|$ folds each antipodal orthant onto the positive one.\nRepeating the classical proof of Wendel’s theorem shows that\n\\[\n\\mathscr E_{d}\n   \\iff \\ker P\\cap S^{\\,n-1}\\ \\text{contains a vector with all\n        coordinates of the same sign}.\n\\]\nConditioned on $\\mathscr E_{d}$ the unique such vector in\n$\\ker P\\cap S^{\\,n-1}$ is therefore uniformly distributed on\n$S^{\\,n-1}_{+}$.  Consequently\n\\[\n\\bigl(\\Lambda_{0},\\dots ,\\Lambda_{d+1}\\bigr)\n  =\\frac{W}{\\sum_{i=1}^{n}W_{i}}\n  =\\frac{H_{1},\\dots ,H_{n}}{\\sum_{i=1}^{n}H_{i}}\n  \\qquad \\text{(under the conditioning } \\mathscr E_{d}\\text{).}\n\\tag{1}\n\\]\n\n\\emph{Step 2: The transformation to $(t,\\lambda)$–coordinates.}\nSet\n\\[\nS_{n}:=\\sum_{j=1}^{n}H_{j},\\qquad\n\\Lambda_{i}:=\\frac{H_{i}}{S_{n}},\\qquad\nt:=S_{n},\\qquad\n\\lambda:=(\\Lambda_{1},\\dots ,\\Lambda_{n}).\n\\]\nThe map\n\\[\n\\Phi\\colon (0,\\infty)\\times\\Delta_{n-1}\\longrightarrow (0,\\infty)^{n},\n\\qquad\n\\Phi(t,\\lambda)=\\bigl(t\\lambda_{1},\\dots ,t\\lambda_{n}\\bigr),\n\\]\nhas Jacobian determinant $t^{\\,n-1}$.  Since the $H_{i}$ are\nindependent with density\n$p(h)=\\sqrt{2/\\pi}\\,\\mathrm e^{-h^{2}/2}\\mathbf 1_{(0,\\infty)}(h)$,\nthe joint density of $(t,\\lambda)$ equals\n\\[\ng(t,\\lambda)=\n  t^{\\,n-1}\\bigl(\\tfrac{2}{\\pi}\\bigr)^{\\!n/2}\n  \\exp\\!\\Bigl\\{-\\tfrac{t^{2}}{2}\\sum_{i=1}^{n}\\lambda_{i}^{2}\\Bigr\\},\n  \\qquad t>0,\\;\\lambda\\in\\Delta_{n-1}.\n\\]\n\n\\emph{Step 3: Marginalising out $t$.}\nLet $A(\\lambda):=\\sum_{i=1}^{n}\\lambda_{i}^{2}$.  Using the\nGamma–integral\n\\[\n\\int_{0}^{\\infty}t^{\\,n-1}\\mathrm e^{-\\beta t^{2}}\\,\\mathrm dt\n  =\\frac{1}{2}\\beta^{-n/2}\\Gamma\\!\\bigl(\\tfrac{n}{2}\\bigr)\n  \\qquad(\\beta>0),\n\\]\nwe obtain the marginal density of $\\Lambda$:\n\\[\nf_{\\Lambda}(\\lambda)=\n  (\\tfrac{2}{\\pi})^{\\!n/2}\\,2^{\\frac{n}{2}-1}\n  \\Gamma\\!\\bigl(\\tfrac{n}{2}\\bigr)\\,\n  A(\\lambda)^{-\\,n/2},\n  \\qquad\\lambda\\in\\Delta_{n-1}.\n\\]\nNormalising over $\\Delta_{n-1}$ furnishes\n\\[\nC_{d}=\n\\frac{2^{\\frac{d}{2}}\\Gamma\\!\\bigl(\\tfrac{d+2}{2}\\bigr)\n      \\bigl(\\tfrac{2}{\\pi}\\bigr)^{\\frac{d+2}{2}}}\n     {\\displaystyle\n      \\int_{\\Delta_{d+1}}\n          \\bigl(x_{0}^{2}+\\dots +x_{d+1}^{2}\\bigr)^{-\\frac{d+2}{2}}\n          \\,\\mathrm dx},\n\\]\nand completes the proof of the claimed density.  The integrand depends\non $\\lambda$ only via $\\lVert\\lambda\\rVert_{2}$, hence the law is\nsymmetric–radial.\n\n\\bigskip\n{\\bf (c) Asymptotics of $M_{d}$.}\n\nRelation (1) gives\n\\[\nM_{d}\n   =\\frac{\\max_{1\\le i\\le n}H_{i}}{\\sum_{j=1}^{n}H_{j}}\n   \\qquad\\bigl(n=d+2,\\;\n          H_{j}=|Z_{j}|,\\;\n          Z_{j}\\sim N(0,1)\\bigr).\n\\tag{2}\n\\]\n\n\\emph{Upper bound.}\nPut $S_{n}:=\\sum_{j=1}^{n}H_{j}$ and $\\mu:=\\mathbb E[H_{1}]\n       =\\sqrt{2/\\pi}$.\nA Chernoff bound yields\n$\\mathbb P\\!\\bigl(S_{n}\\le (\\mu/2)\\,n\\bigr)\\le\\mathrm e^{-c_{0}n}$ for\nsome universal $c_{0}>0$.  Conditioning on the complement event and\nusing $\\mathbb E[\\max_{j}H_{j}]\\le\\sqrt{2\\log n}+C_{0}$ (Gaussian\nmaximum bound) gives\n\\[\n\\mathbb E[M_{d}]\n   \\le\n   \\frac{\\sqrt{2\\log n}+C_{0}}{(\\mu/2)\\,n}\n      +\\frac{\\mu}{2}\\,\\mathrm e^{-c_{0}n}\n   \\le\n   \\frac{c_{2}\\sqrt{\\log d}}{d},\n\\]\nfor a universal constant $c_{2}$.\n\n\\emph{Lower bound.}\nLet\n$b_{n}:=\\sqrt{2\\log n}-\\frac{\\log\\log n}{\\sqrt{2\\log n}}$;\nstandard extreme–value estimates give\n$\\mathbb P\\!\\bigl(\\max_{j}H_{j}\\ge b_{n}\\bigr)\\ge 3/4$ for $n\\ge 3$.\nA second Chernoff bound\n$\\mathbb P\\!\\bigl(S_{n}\\le (3\\mu/2)\\,n\\bigr)\\ge 3/4$ provides, by the\nunion bound, that with probability at least $1/2$\n\\[\nM_{d}\\ge \\frac{b_{n}}{(3\\mu/2)\\,n}\n       \\ge c_{1}\\frac{\\sqrt{\\log n}}{n},\n\\]\nwhence $\\mathbb E[M_{d}]\\ge c_{1}\\sqrt{\\log d}/d$ for a universal\n$c_{1}>0$.\n\nCombining the upper and lower bounds yields the announced\n$\\Theta\\!\\bigl(\\sqrt{\\log d}\\,/\\,d\\bigr)$ behaviour of $\\mathbb E[M_{d}]$.\n\\hfill$\\square$\n\n\\bigskip",
      "metadata": {
        "replaced_from": "harder_variant",
        "replacement_date": "2025-07-14T01:37:45.563045",
        "was_fixed": false,
        "difficulty_analysis": "1.  Higher dimension and more variables – the problem is stated for arbitrary d ≥ 4 and involves the full (d+1)-tuple of barycentric coordinates, not just the yes/no event of containment.\n\n2.  Additional concepts – beyond probability of containment, parts (b) and (c) demand identification of an entire **probability distribution** (Dirichlet with parameter ½) and the evaluation of a non-trivial statistic (the expected maximum).\n\n3.  Deeper theoretical tools – the solution uses  \n  • linear-algebraic characterisation of convex containment,  \n  • orthogonal invariance and random-matrix reasoning,  \n  • relations between the uniform sphere, Gaussian vectors, chi-square laws, and Dirichlet distributions,  \n  • inclusion–exclusion on the probability simplex, incomplete Beta functions and hypergeometric identities.\n\n4.  Multiple interacting ideas – the chain “kernel of a random matrix → sign pattern → conditional Dirichlet law → order statistics of Dirichlet variables” forces the solver to blend geometry, probability, and special-function calculus.\n\nThese layers of technical sophistication go well beyond both the original problem (a single probability 1/8) and the current kernel variant (probability  that 0 lies in a 4-simplex), fulfilling the requirement of a **significantly harder** kernel variant."
      }
    }
  },
  "checked": true,
  "problem_type": "calculation",
  "iteratively_fixed": true
}