{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,1]],"date-time":"2026-04-01T10:45:03Z","timestamp":1775040303260,"version":"3.50.1"},"reference-count":36,"publisher":"Springer Science and Business Media LLC","issue":"3","license":[{"start":{"date-parts":[[2020,12,10]],"date-time":"2020-12-10T00:00:00Z","timestamp":1607558400000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2020,12,10]],"date-time":"2020-12-10T00:00:00Z","timestamp":1607558400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"DOI":"10.13039\/501100001659","name":"Deutsche Forschungsgemeinschaft","doi-asserted-by":"publisher","award":["EXC 2044 - 390685587"],"award-info":[{"award-number":["EXC 2044 - 390685587"]}],"id":[{"id":"10.13039\/501100001659","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Discrete Comput Geom"],"published-print":{"date-parts":[[2021,10]]},"abstract":"Abstract<\/jats:title>Consider a random simplex $$[X_1,\\ldots ,X_n]$$<\/jats:tex-math>\n \n [<\/mml:mo>\n \n X<\/mml:mi>\n 1<\/mml:mn>\n <\/mml:msub>\n ,<\/mml:mo>\n \u2026<\/mml:mo>\n ,<\/mml:mo>\n \n X<\/mml:mi>\n n<\/mml:mi>\n <\/mml:msub>\n ]<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> defined as the convex hull of independent identically distributed (i.i.d.) random points $$X_1,\\ldots ,X_n$$<\/jats:tex-math>\n \n \n X<\/mml:mi>\n 1<\/mml:mn>\n <\/mml:msub>\n ,<\/mml:mo>\n \u2026<\/mml:mo>\n ,<\/mml:mo>\n \n X<\/mml:mi>\n n<\/mml:mi>\n <\/mml:msub>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> in $$\\mathbb {R}^{n-1}$$<\/jats:tex-math>\n \n \n R<\/mml:mi>\n <\/mml:mrow>\n \n n<\/mml:mi>\n -<\/mml:mo>\n 1<\/mml:mn>\n <\/mml:mrow>\n <\/mml:msup>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> with the following beta density: \"Equation missing\" Let $$J_{n,k}(\\beta )$$<\/jats:tex-math>\n \n \n J<\/mml:mi>\n \n n<\/mml:mi>\n ,<\/mml:mo>\n k<\/mml:mi>\n <\/mml:mrow>\n <\/mml:msub>\n \n (<\/mml:mo>\n \u03b2<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> be the expected internal angle of the simplex $$[X_1,\\ldots ,X_n]$$<\/jats:tex-math>\n \n [<\/mml:mo>\n \n X<\/mml:mi>\n 1<\/mml:mn>\n <\/mml:msub>\n ,<\/mml:mo>\n \u2026<\/mml:mo>\n ,<\/mml:mo>\n \n X<\/mml:mi>\n n<\/mml:mi>\n <\/mml:msub>\n ]<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> at its face $$[X_1,\\ldots ,X_k]$$<\/jats:tex-math>\n \n [<\/mml:mo>\n \n X<\/mml:mi>\n 1<\/mml:mn>\n <\/mml:msub>\n ,<\/mml:mo>\n \u2026<\/mml:mo>\n ,<\/mml:mo>\n \n X<\/mml:mi>\n k<\/mml:mi>\n <\/mml:msub>\n ]<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>. Define $${\\tilde{J}}_{n,k}(\\beta )$$<\/jats:tex-math>\n \n \n \n J<\/mml:mi>\n ~<\/mml:mo>\n <\/mml:mover>\n \n n<\/mml:mi>\n ,<\/mml:mo>\n k<\/mml:mi>\n <\/mml:mrow>\n <\/mml:msub>\n \n (<\/mml:mo>\n \u03b2<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> analogously for i.i.d. random points distributed according to the beta$$'$$<\/jats:tex-math>\n \n \n \u2032<\/mml:mo>\n <\/mml:msup>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> density $${\\tilde{f}}_{n-1,\\beta } (x) \\propto (1+\\Vert x\\Vert ^2)^{-\\beta }, x\\in \\mathbb {R}^{n-1}, \\beta > ({n-1})\/{2}.$$<\/jats:tex-math>\n \n \n \n f<\/mml:mi>\n ~<\/mml:mo>\n <\/mml:mover>\n \n n<\/mml:mi>\n -<\/mml:mo>\n 1<\/mml:mn>\n ,<\/mml:mo>\n \u03b2<\/mml:mi>\n <\/mml:mrow>\n <\/mml:msub>\n \n (<\/mml:mo>\n x<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n \u221d<\/mml:mo>\n \n \n (<\/mml:mo>\n 1<\/mml:mn>\n +<\/mml:mo>\n \u2016<\/mml:mo>\n x<\/mml:mi>\n \u2016<\/mml:mo>\n <\/mml:mrow>\n 2<\/mml:mn>\n <\/mml:msup>\n \n \n )<\/mml:mo>\n <\/mml:mrow>\n \n -<\/mml:mo>\n \u03b2<\/mml:mi>\n <\/mml:mrow>\n <\/mml:msup>\n ,<\/mml:mo>\n x<\/mml:mi>\n \u2208<\/mml:mo>\n \n \n R<\/mml:mi>\n <\/mml:mrow>\n \n n<\/mml:mi>\n -<\/mml:mo>\n 1<\/mml:mn>\n <\/mml:mrow>\n <\/mml:msup>\n ,<\/mml:mo>\n \u03b2<\/mml:mi>\n ><\/mml:mo>\n \n (<\/mml:mo>\n \n n<\/mml:mi>\n -<\/mml:mo>\n 1<\/mml:mn>\n <\/mml:mrow>\n )<\/mml:mo>\n <\/mml:mrow>\n \/<\/mml:mo>\n 2<\/mml:mn>\n .<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> We derive formulae for $$J_{n,k}(\\beta )$$<\/jats:tex-math>\n \n \n J<\/mml:mi>\n \n n<\/mml:mi>\n ,<\/mml:mo>\n k<\/mml:mi>\n <\/mml:mrow>\n <\/mml:msub>\n \n (<\/mml:mo>\n \u03b2<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> and $${\\tilde{J}}_{n,k}(\\beta )$$<\/jats:tex-math>\n \n \n \n J<\/mml:mi>\n ~<\/mml:mo>\n <\/mml:mover>\n \n n<\/mml:mi>\n ,<\/mml:mo>\n k<\/mml:mi>\n <\/mml:mrow>\n <\/mml:msub>\n \n (<\/mml:mo>\n \u03b2<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> which make it possible to compute these quantities symbolically, in finitely many steps, for any integer or half-integer value of\u00a0$$\\beta $$<\/jats:tex-math>\n \u03b2<\/mml:mi>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>. For $$J_{n,1}(\\pm 1\/2)$$<\/jats:tex-math>\n \n \n J<\/mml:mi>\n \n n<\/mml:mi>\n ,<\/mml:mo>\n 1<\/mml:mn>\n <\/mml:mrow>\n <\/mml:msub>\n \n (<\/mml:mo>\n \u00b1<\/mml:mo>\n 1<\/mml:mn>\n \/<\/mml:mo>\n 2<\/mml:mn>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> we even provide explicit formulae in terms of products of Gamma functions. We give applications of these results to two seemingly unrelated problems of stochastic geometry: (i)\u00a0We compute explicitly the expected f<\/jats:italic>-vectors of the typical Poisson\u2013Voronoi cells in dimensions up to\u00a010. (ii)\u00a0Consider the random polytope $$K_{n,d} := [U_1,\\ldots ,U_n]$$<\/jats:tex-math>\n \n \n K<\/mml:mi>\n \n n<\/mml:mi>\n ,<\/mml:mo>\n d<\/mml:mi>\n <\/mml:mrow>\n <\/mml:msub>\n :<\/mml:mo>\n =<\/mml:mo>\n \n [<\/mml:mo>\n \n U<\/mml:mi>\n 1<\/mml:mn>\n <\/mml:msub>\n ,<\/mml:mo>\n \u2026<\/mml:mo>\n ,<\/mml:mo>\n \n U<\/mml:mi>\n n<\/mml:mi>\n <\/mml:msub>\n ]<\/mml:mo>\n <\/mml:mrow>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> where $$U_1,\\ldots ,U_n$$<\/jats:tex-math>\n \n \n U<\/mml:mi>\n 1<\/mml:mn>\n <\/mml:msub>\n ,<\/mml:mo>\n \u2026<\/mml:mo>\n ,<\/mml:mo>\n \n U<\/mml:mi>\n n<\/mml:mi>\n <\/mml:msub>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> are i.i.d. random points sampled uniformly inside some d<\/jats:italic>-dimensional convex body K<\/jats:italic> with smooth boundary and unit volume. Reitzner (Adv. Math. 191<\/jats:bold>(1), 178\u2013208 (2005)) proved the existence of the limit of the normalised expected f<\/jats:italic>-vector of\u00a0$$K_{n,d}$$<\/jats:tex-math>\n \n K<\/mml:mi>\n \n n<\/mml:mi>\n ,<\/mml:mo>\n d<\/mml:mi>\n <\/mml:mrow>\n <\/mml:msub>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>: $$ \\lim _{n\\rightarrow \\infty } n^{-{({d-1})\/({d+1})}}{\\mathbb {E}}{\\mathbf {f}}(K_{n,d}) = {\\mathbf {c}}_d \\cdot \\Omega (K),$$<\/jats:tex-math>\n \n \n lim<\/mml:mo>\n \n n<\/mml:mi>\n \u2192<\/mml:mo>\n \u221e<\/mml:mi>\n <\/mml:mrow>\n <\/mml:msub>\n \n n<\/mml:mi>\n \n -<\/mml:mo>\n \n (<\/mml:mo>\n \n d<\/mml:mi>\n -<\/mml:mo>\n 1<\/mml:mn>\n <\/mml:mrow>\n )<\/mml:mo>\n \/<\/mml:mo>\n (<\/mml:mo>\n \n d<\/mml:mi>\n +<\/mml:mo>\n 1<\/mml:mn>\n <\/mml:mrow>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:mrow>\n <\/mml:msup>\n E<\/mml:mi>\n f<\/mml:mi>\n \n (<\/mml:mo>\n \n K<\/mml:mi>\n \n n<\/mml:mi>\n ,<\/mml:mo>\n d<\/mml:mi>\n <\/mml:mrow>\n <\/mml:msub>\n )<\/mml:mo>\n <\/mml:mrow>\n =<\/mml:mo>\n \n c<\/mml:mi>\n d<\/mml:mi>\n <\/mml:msub>\n \u00b7<\/mml:mo>\n \u03a9<\/mml:mi>\n \n (<\/mml:mo>\n K<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n ,<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> where $$\\Omega (K)$$<\/jats:tex-math>\n \n \u03a9<\/mml:mi>\n (<\/mml:mo>\n K<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> is the affine surface area of\u00a0K<\/jats:italic>, and $${\\mathbf {c}}_d$$<\/jats:tex-math>\n \n c<\/mml:mi>\n d<\/mml:mi>\n <\/mml:msub>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> is an unknown vector not depending on K<\/jats:italic>. We compute $${\\mathbf {c}}_d$$<\/jats:tex-math>\n \n c<\/mml:mi>\n d<\/mml:mi>\n <\/mml:msub>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> explicitly in dimensions up to $$d=10$$<\/jats:tex-math>\n \n d<\/mml:mi>\n =<\/mml:mo>\n 10<\/mml:mn>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> and also solve the analogous problem for random polytopes with vertices distributed uniformly on the sphere.<\/jats:p>","DOI":"10.1007\/s00454-020-00259-z","type":"journal-article","created":{"date-parts":[[2020,12,10]],"date-time":"2020-12-10T18:05:14Z","timestamp":1607623514000},"page":"902-937","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":13,"title":["Recursive Scheme for Angles of Random Simplices, and Applications to Random Polytopes"],"prefix":"10.1007","volume":"66","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-8483-3373","authenticated-orcid":false,"given":"Zakhar","family":"Kabluchko","sequence":"first","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2020,12,10]]},"reference":[{"issue":"3","key":"259_CR1","doi-asserted-by":"publisher","first-page":"277","DOI":"10.1111\/j.1365-2818.1988.tb04688.x","volume":"151","author":"F Affentranger","year":"1988","unstructured":"Affentranger, F.: The expected volume of a random polytope in a ball. 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