# Department Seminars & Colloquia

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ZOOM Meeting ID: 868 7549 9085 Direct link: https://kaist.zoom.us/j/86875499085

ZOOM Meeting ID: 868 7549 9085 Direct link: https://kaist.zoom.us/j/86875499085

Helly-type theorems and problems form a nice area of discrete geometry. I will start with the notable theorems of Radon and Tverberg and mention the following conjectural extension.

For a set *X* of points *x(1), x(2),...,x(n)* in some real vector space *V* we denote by *T(X,r)* the set of points in *X* that belong to the convex hulls of r pairwise disjoint subsets of *X*.

We let
*t(X,r)* = 1 + dim(*T(X,r)*).

Radon's theorem asserts that

If *t(X,1)* < |*X*| then *t(X, 2)* > 0.

If

*t(X,1)*+

*t(X,2)*< |

*X*| then

*t(X,3)*>0.

In the lecture I will discuss connections with topology and with various problems in graph theory.

I will also mention questions regarding dimensions of intersection of convex sets.

1) A lecture (from 1999): An invitation to Tverberg Theorem: https://youtu.be/Wjg1_QwjUos

2) A paper on Helly type problems by Barany and me https://arxiv.org/abs/2108.08804

3) A link to Barany's book: Combinatorial convexity https://www.amazon.com/Combinatorial-Convexity-University-Lecture-77/dp/1470467097

ZOOM Meeting ID: 868 7549 9085 Direct link: https://kaist.zoom.us/j/86875499085