Soumi Nandi: Colorful Helly theorem for piercing boxes with two points

Description of video

Date: 9/30/22
Speaker :Nandi Soumi

For any natural number n, a family F of subsets of a space X is said to be {\em n-pierceable}, if there exists AX with |A|n such that for any FFFA.
Helly's theorem, one of the fundamental results in discrete geometry, says that for any finite family F of convex sets in Rd, if every (d+1)-tuple from F is 1-pierceable, then the whole family F is 1-pierceable. Unfortunately, for n2, a similar statement about the n-pierceable sets is not valid for general convex sets. Danzer and Grünbaum proved the first and one of the most important Helly type results on multi-pierceable families; viz. famlies of axis parallel boxes.
One important generalization of Helly's theorem is Colorful Helly's Theorem. In this talk, we shall prove a colorful version of Danzer and Grünbaum's 2-pierceability result for families of axis parallel boxes. This work was jointly done with Sourav Chakraborty and Arijit Ghosh.