Geometry Seminar

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On spheres with k points inside

Speaker: Alexey Garber, The University of Texas Rio Grande Valley

Location: Warren Weaver Hall 1314 in person and on Zoom

Date: Tuesday, April 8, 2025, 6 p.m.

Synopsis:

A classical result of Delone (Delaunay) claims that for a finite and generic point set \(A\) in \(\mathbb{R}^d\), every generic point in the convex hull of \(A\) belongs to exactly one simplex with empty circumsphere. The collection of all these simplices is called the Delaunay triangulation of \(A\). In the talk I will discuss a generalization of Delaunay’s result to the case of simplices with \(k\) points inside their circumspheres. I will also talk about possible extensions to the case of weighted points sets and point sets in \(\mathbb{S}^d\), and sketch a few combinatorial and geometric consequences related to \(k\)-sets and hypersimplices.

The talk is based on joint work with Herbert Edelsbrunner and Morteza Saghafian.

Notes:

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