August 2023 Angle Sums of Random Polytopes
Thomas Godland, Zakhar Kabluchko, Dmitry Zaporozhets
Michigan Math. J. 73(4): 815-842 (August 2023). DOI: 10.1307/mmj/20206021


For two families of random polytopes, we compute explicitly the expected sums of the conic intrinsic volumes and the Grassmann angles at all faces of any given dimension of the polytope under consideration. As particular cases, we compute the expected sums of internal and external angles at all faces of any fixed dimension. The first family is the Gaussian polytopes defined as convex hulls of i.i.d. samples from a nondegenerate Gaussian distribution in Rd. The second family is convex hulls of random walks with exchangeable increments satisfying certain mild general position assumption. The expected sums are expressed in terms of the angles of the regular simplices and the Stirling numbers, respectively. There are nontrivial analogies between these two settings. Further, we compute the angle sums for Gaussian projections of arbitrary polyhedral sets, of which the Gaussian polytopes are a particular case. Also, we show that the expected Grassmann angle sums of a random polytope with a rotationally invariant law are invariant under affine transformations. Of independent interest may be also results on the faces of linear images of polyhedral sets. These results are well known, but it seems that no detailed proofs can be found in the existing literature.


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Thomas Godland. Zakhar Kabluchko. Dmitry Zaporozhets. "Angle Sums of Random Polytopes." Michigan Math. J. 73 (4) 815 - 842, August 2023.


Received: 29 December 2020; Revised: 12 August 2021; Published: August 2023
First available in Project Euclid: 31 August 2023

MathSciNet: MR4634982
Digital Object Identifier: 10.1307/mmj/20206021

Keywords: 51F15 , 52A22 , 52A55 , 60D05

Rights: Copyright © 2023 The University of Michigan


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Vol.73 • No. 4 • August 2023
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