Random triangles and random inscribed polytopes

Herbert Edelsbrunner

Given three random points on a circle, the triangle they form is acute with probability $1/4$. In contrast, the triangle formed by three random points in the 2-sphere is acute with probability $1/2$. 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 $1/2$. Picking $n$ 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 $2 (n-1)/(n+1)$, $4\pi [(n-1)(n-2)]/[(n+1)(n+2)]$, and $(4\pi/3) [(n-1)(n-2)(n-3)]/[(n+1)(n+2)(n+3)]$. These formulas are not new but our combinatorial proofs are. Work with Arseniy Akopyan and Anton Nikitenko.

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