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Abstract:

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

Registro:

Documento: Artículo
Título:The expected volume of a random polytope in a ball
Autor:Affentranger, F.
Filiación:Departamento de Matemática, Universidad de Buenos Aires, Ciudad Universitaria, Buenos Aires, 1428, Argentina
Palabras clave: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
Año:1988
Volumen:151
Número:3
Página de inicio:277
Página de fin:287
DOI: http://dx.doi.org/10.1111/j.1365-2818.1988.tb04688.x
Título revista:Journal of Microscopy
Título revista abreviado:J. Microsc.
ISSN:00222720
Registro:https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_00222720_v151_n3_p277_Affentranger

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Citas:

---------- APA ----------
(1988) . The expected volume of a random polytope in a ball. Journal of Microscopy, 151(3), 277-287.
http://dx.doi.org/10.1111/j.1365-2818.1988.tb04688.x
---------- CHICAGO ----------
Affentranger, F. "The expected volume of a random polytope in a ball" . Journal of Microscopy 151, no. 3 (1988) : 277-287.
http://dx.doi.org/10.1111/j.1365-2818.1988.tb04688.x
---------- MLA ----------
Affentranger, F. "The expected volume of a random polytope in a ball" . Journal of Microscopy, vol. 151, no. 3, 1988, pp. 277-287.
http://dx.doi.org/10.1111/j.1365-2818.1988.tb04688.x
---------- VANCOUVER ----------
Affentranger, F. The expected volume of a random polytope in a ball. J. Microsc. 1988;151(3):277-287.
http://dx.doi.org/10.1111/j.1365-2818.1988.tb04688.x