On Expected Face Numbers of Random Beta and Beta' Polytopes

DOI10.1007/S13366-022-00626-2zbMATH Open1509.52005arXiv2107.06655MaRDI QIDQ6372783

Zakhar Kabluchko

Publication date: 14 July 2021

Abstract: The random beta polytope is defined as the convex hull of $n$ independent random points with the density proportional to $$ on the $d$-dimensional unit ball, where $$ is a parameter. Similarly, the random beta' polytope is defined as the convex hull of $n$ independent random points with the density proportional to $$ on $mathbb{R}^d$, where $$. In a previous work [Angles of random simplices and face numbers of random polytopes, Adv. Math., 380 (2021), 107612], we established exact and explicit formulae for the expected $f$-vectors of these random polytopes in terms of certain definite integrals. In the present paper, we use purely algebraic manipulations to derive several identities for these integrals which yield alternative formulae for the expected $f$-vectors. Similar algebraic manipulations apply to Stirling numbers and yield the following identity: ${\displaystyle }$ sum_{s=0}^k genfrac{{}{}}{0pt}{}{n-s}{d-s} (d-s) genfrac{[}{]}{0pt}{}{d-s}{k-s} = sum_{s=0}^k (-1)^s genfrac{{}{}}{0pt}{}{n-s}{d} genfrac{[}{]}{0pt}{}{d+1}{k-s} = sum_{s=0}^{d-k} (-1)^s genfrac{{}{}}{0pt}{}{n+1}{d-s} genfrac{[}{]}{0pt}{}{d-s}{k}. ${\displaystyle }$

Mathematics Subject Classification ID

Geometric probability and stochastic geometry (60D05) Bell and Stirling numbers (11B73) Combinatorial properties of polytopes and polyhedra (number of faces, shortest paths, etc.) (52B05) (n)-dimensional polytopes (52B11) Recurrences (11B37) Other functions defined by series and integrals (33E20) Other functions coming from differential, difference and integral equations (33E30) Spherical and hyperbolic convexity (52A55) Approximation by convex sets (52A27) Random convex sets and integral geometry (aspects of convex geometry) (52A22)

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