BCTCS 2021

Invited Talk
Algorithms

Random triangles and random inscribed polytopes

Herbert Edelsbrunner

on  Mon, 14:00 ! Live for  60min

Given three random points on a circle, the triangle they form is acute with probability 1/41/4. In contrast, the triangle formed by three random points in the 2-sphere is acute with probability 1/21/2. Both of these claims have short geometric proofs. We use the latter fact to prove that a triangle in the boundary of a random inscribed 3-polytope is acute with probability 1/21/2. Picking nn points uniformly at random on the 2-sphere, we take the convex hull, which is an inscribed 3-polytope. The expected mean width, surface area, and volume of this polytope are 2(n1)/(n+1)2 (n-1)/(n+1), 4π[(n1)(n2)]/[(n+1)(n+2)]4\pi [(n-1)(n-2)]/[(n+1)(n+2)], and (4π/3)[(n1)(n2)(n3)]/[(n+1)(n+2)(n+3)](4\pi/3) [(n-1)(n-2)(n-3)]/[(n+1)(n+2)(n+3)]. These formulas are not new but our combinatorial proofs are. Work with Arseniy Akopyan and Anton Nikitenko.

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