Stabbing boxes with finitely many axis-parallel lines and flats

Sutanoya Chakraborty, Arijit Ghosh, Soumi Nandi

Abstract: In this short note, we provide the necessary and sufficient condition for an infinite collection of axis-parallel boxes in Rd<math><msup is="true"><mrow is="true"><mi mathvariant="double-struck" is="true">R</mi></mrow><mrow is="true"><mi is="true">d</mi></mrow></msup></math> to be pierceable by finitely many axis-parallel k-flats, where 0≤k<d<math><mn is="true">0</mn><mo is="true">≤</mo><mi is="true">k</mi><mo linebreak="goodbreak" linebreakstyle="after" is="true">&lt;</mo><mi is="true">d</mi></math>. We also consider colorful generalizations of the above result and establish their feasibility. The problem considered in this paper is an infinite variant of the Hadwiger-Debrunner (p,q)<math><mo stretchy="false" is="true">(</mo><mi is="true">p</mi><mo is="true">,</mo><mi is="true">q</mi><mo stretchy="false" is="true">)</mo></math>-problem.