Lower Bounds for Convexity Testing

Xi Chen, Anindya De, Shivam Nadimpalli, Rocco A. Servedio, Erik Waingarten

Abstract: We consider the problem of testing whether an unknown and arbitrary set S ⊆ ℝn (given as a black-box membership oracle) is convex, versus ε-far from every convex set, under the standard Gaussian distribution.The current state-of-the-art testing algorithms for this problem make non-adaptive queries, both for the standard testing problem and for tolerant testing.We give the first lower bounds for convexity testing in the black-box query model:• We show that any one-sided tester (which may be adaptive) must use at least nΩ(1) queries in order to test to some constant accuracy ε > 0.• We show that any non-adaptive tolerant tester (which may make two-sided errors) must use at least 2Ω(n1/4) queries to distinguish sets that are ε1-close to convex versus ε2-far from convex, for some absolute constants 0 < ε1 < ε2.Finally, we also show that for any constant c > 0, any non-adaptive tester (which may make two-sided errors) must use at least n 1/4-c queries in order to test to some constant accuracy ε > 0.

External IDs:dblp:conf/soda/0001DNSW25