Counting the Shape of a Drum

✍ Scribed by Yu. Baryshnikov

• Publisher: Elsevier Science
• Year: 1996
• Tongue: English
• Weight: 272 KB
• Volume: 17
• Category: Article
• ISSN: 0196-8858
• DOI: 10.1006/aama.1996.0005

✦ Synopsis

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 in¨erse problemᎏwhether 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 Ž is a triangle for m s 4, 5, . . . determine generic polygon P unambiguously up to . affine transformations . Some general constructions, conjectures, and corollaries are presented. ᮊ 1996 Academic Press, Inc. 0 volume 2-form. By random points in P we will understand points X , X , . . . 1 2 distributed independently identically and uniformly in P.

A great many of information has been gathered about the finite samples Ä 4 X sX , X, . . . , X of random points in P. A majority of the results 1 2 m addresses the most intuitive object associated to the random sample X, its convex hull. The average number of vertices of this new random polygon inside P is known to grow, up to a universal multiplicative constant, as n log m, n s ࠻P and its asymptotic distribution is normal with the vari-0 ance of the same order of magnitude as its average. The question about the average area of the convex hull of randomly chosen three points in P is

The author was supported by DFG.