Counting the shape of a drum

Autor(en): Baryshnikov, Y
Stichwörter: Mathematics; Mathematics, Applied
Erscheinungsdatum: 1996
Herausgeber: ACADEMIC PRESS INC JNL-COMP SUBSCRIPTIONS
Journal: ADVANCES IN APPLIED MATHEMATICS
Volumen: 17
Ausgabe: 1
Startseite: 101
Seitenende: 116
Zusammenfassung: 
Let P be a convex polygon. A great many papers are devoted to investigation of various random values associated to the finite samples from the uniform distribution in the polygon, such as the number of the vertices of the convex hull of the sample, its circumference, probability that th convex hull has the fixed number of vertices, etc. In this note we address an inverse problem-whether and to what extent the distributions of these random values determine P. We show that Sylvester numbers, that is, the probabilities that the convex hull of m random point is a triangle for m = 4,5,... determine generic polygon P unambiguously (up to affine transformations). Some general constructions, conjectures, and corollaries are presented. (C) 1996 Academic Press, Inc.
ISSN: 01968858
DOI: 10.1006/aama.1996.0005

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