Premium
EXPECTED f ‐VECTOR OF THE POISSON ZERO POLYTOPE AND RANDOM CONVEX HULLS IN THE HALF‐SPHERE
Author(s) -
Kabluchko Zakhar
Publication year - 2020
Publication title -
mathematika
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.955
H-Index - 29
eISSN - 2041-7942
pISSN - 0025-5793
DOI - 10.1112/mtk.12056
Subject(s) - mathematics , combinatorics , polytope , convex hull , simplex , convex polytope , hyperplane , zero (linguistics) , convex body , mixed volume , regular polygon , orthogonal convex hull , mathematical analysis , geometry , convex set , convex optimization , linguistics , philosophy
We prove an explicit combinatorial formula for the expected number of faces of the zero polytope of the homogeneous and isotropic Poisson hyperplane tessellation in R d . The expected f ‐vector is expressed through the coefficients of the polynomial( 1 + ( d − 1 ) 2 x 2 ) ( 1 + ( d − 3 ) 2 x 2 ) ( 1 + ( d − 5 ) 2 x 2 ) … . Also, we compute explicitly the expected f ‐vector and the expected volume of the spherical convex hull of n random points sampled uniformly and independently from the d ‐dimensional half‐sphere. In the case when n = d + 2 , we compute the probability that this spherical convex hull is a spherical simplex, thus solving the half‐sphere analogue of the Sylvester four‐point problem.