The Sylvester (d+2)-points problem deals with the probability S(K) that d+2 random points taken from a convex compact subset K of R^d are not the vertices of any convex polytope and asks for which sets S(K) is minimal or maximal. While it is known that ellipsoids are the only minimizers of S(K) , the problem of the maximum is still open, unless d=2 . In this paper we study generalizations of S(K) , which include the Busemann functional and a functional introduced by Bourgain, Meyer and Pajor in connection with the local theory of Banach spaces. We show that also for these functionals ellipsoids are the only minimizers and for d=2 triangles (or parallelograms, in the symmetric case) are maximizers.

Campi, S., & Gronchi, P. (2006). Extremal convex sets for Sylvester-Busemann type functionals. APPLICABLE ANALYSIS, 85(1-3), 129-141 [10.1080/00036810500277579].

Extremal convex sets for Sylvester-Busemann type functionals

CAMPI, STEFANO;
2006

Abstract

The Sylvester (d+2)-points problem deals with the probability S(K) that d+2 random points taken from a convex compact subset K of R^d are not the vertices of any convex polytope and asks for which sets S(K) is minimal or maximal. While it is known that ellipsoids are the only minimizers of S(K) , the problem of the maximum is still open, unless d=2 . In this paper we study generalizations of S(K) , which include the Busemann functional and a functional introduced by Bourgain, Meyer and Pajor in connection with the local theory of Banach spaces. We show that also for these functionals ellipsoids are the only minimizers and for d=2 triangles (or parallelograms, in the symmetric case) are maximizers.
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Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/11365/22094
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