The expected volume of a random polytope in a ball
For any convex body K in d‐dimensional Euclidean space Ed(d≥2) and for integers n and i, n ≥ d + 1,1 ≤ i ≤ n, let V(d) n‐ii(K) be the expected volume of the convex hull Hn‐i, i of n independent random points, of which n‐i are uniformly distributed in the interior, the other i on the boundary of K. W...
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todo:paper_00222720_v151_n3_p277_Affentranger2023-10-03T14:30:19Z The expected volume of a random polytope in a ball Affentranger, F. Crofton's theorem on mean values expected volume of a random polytope geometric probabilities inscribed random polytopes integral geometry set of uniform random points stochastic geometry Sylvester's problem For any convex body K in d‐dimensional Euclidean space Ed(d≥2) and for integers n and i, n ≥ d + 1,1 ≤ i ≤ n, let V(d) n‐ii(K) be the expected volume of the convex hull Hn‐i, i of n independent random points, of which n‐i are uniformly distributed in the interior, the other i on the boundary of K. We develop an integral formula for V(d) n‐i, i(K) for the case that K is a d‐dimensional unit ball by considering an adequate decomposition of V(d) n‐i, i into d‐dimensional simplices. To solve the important case i = 0, that is the case in which all points are chosen at random from the interior of Bd, we require in addition Crofton's theorem on mean values. We illustrate the usefulness of our results by treating some special cases and by giving numerical values for the planar and the three‐dimensional cases. 1988 Blackwell Science Ltd JOUR info:eu-repo/semantics/openAccess http://creativecommons.org/licenses/by/2.5/ar http://hdl.handle.net/20.500.12110/paper_00222720_v151_n3_p277_Affentranger |
| institution |
Universidad de Buenos Aires |
| institution_str |
I-28 |
| repository_str |
R-134 |
| collection |
Biblioteca Digital - Facultad de Ciencias Exactas y Naturales (UBA) |
| topic |
Crofton's theorem on mean values expected volume of a random polytope geometric probabilities inscribed random polytopes integral geometry set of uniform random points stochastic geometry Sylvester's problem |
| spellingShingle |
Crofton's theorem on mean values expected volume of a random polytope geometric probabilities inscribed random polytopes integral geometry set of uniform random points stochastic geometry Sylvester's problem Affentranger, F. The expected volume of a random polytope in a ball |
| topic_facet |
Crofton's theorem on mean values expected volume of a random polytope geometric probabilities inscribed random polytopes integral geometry set of uniform random points stochastic geometry Sylvester's problem |
| description |
For any convex body K in d‐dimensional Euclidean space Ed(d≥2) and for integers n and i, n ≥ d + 1,1 ≤ i ≤ n, let V(d) n‐ii(K) be the expected volume of the convex hull Hn‐i, i of n independent random points, of which n‐i are uniformly distributed in the interior, the other i on the boundary of K. We develop an integral formula for V(d) n‐i, i(K) for the case that K is a d‐dimensional unit ball by considering an adequate decomposition of V(d) n‐i, i into d‐dimensional simplices. To solve the important case i = 0, that is the case in which all points are chosen at random from the interior of Bd, we require in addition Crofton's theorem on mean values. We illustrate the usefulness of our results by treating some special cases and by giving numerical values for the planar and the three‐dimensional cases. 1988 Blackwell Science Ltd |
| format |
JOUR |
| author |
Affentranger, F. |
| author_facet |
Affentranger, F. |
| author_sort |
Affentranger, F. |
| title |
The expected volume of a random polytope in a ball |
| title_short |
The expected volume of a random polytope in a ball |
| title_full |
The expected volume of a random polytope in a ball |
| title_fullStr |
The expected volume of a random polytope in a ball |
| title_full_unstemmed |
The expected volume of a random polytope in a ball |
| title_sort |
expected volume of a random polytope in a ball |
| url |
http://hdl.handle.net/20.500.12110/paper_00222720_v151_n3_p277_Affentranger |
| work_keys_str_mv |
AT affentrangerf theexpectedvolumeofarandompolytopeinaball AT affentrangerf expectedvolumeofarandompolytopeinaball |
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