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Fodor Ferenc (SZTE): Random approximations of convex bodies by balls

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Wednesday, 21. March 2018, 14:00 - 16:00

Abstract: Approximation of convex bodies by random polytopes is one of the oldest topics of geometric probability. Its origins go back to the famous four-point problem of Sylvester in the 1860s. The most frequently investigated model of random polytopes is when one takes the convex hull of n i.i.d. uniform random points in a d-dimensional convex body K. Certain geometric quantities of the random polytopes, such as volume, surface area, etc, approach those of K with high probability as n tends to infinity. It is one of the important questions how fast this convergence is and how it depends on the facial structure of K.

In this talk we will consider a variant of the uniform model in which we generate the random objects (called random ball-polytopes) by the intersection of equal radius closed balls instead of closed half-spaces. We will primarily investigate the asymptotic behaviour of the number of facets of random ball-polytopes in smooth convex bodies as n tends to infinity. We will generalize some earlier joint results with P. Kevei and V. Vigh from the plane to arbitrary dimensions. We will prove the interesting phenomenon that the expected number of proper facets of uniform random ball-polytopes in a ball tends to a constant that depends only on the dimension.
Location : Szeged, Aradi vértanúk tere 1., Riesz terem.


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