BIRS Workshop Lecture Videos

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BIRS Workshop Lecture Videos

Floating Bodies and Random Approximation Besau, Florian


It is a classic result that the expected volume difference between a convex body and a random polytope, i.e., the convex hull of i.i.d. random points chosen uniformly from the convex body, converges to the affine surface area of the convex body as the number of points goes to infinity. Furthermore, if the convex body is actually a polytope, then the affine surface area vanishes and the first term in the asymptotic expansion of the volume difference depends on the number of (complete) flags of the polytope. Remarkably, a similar behavior is exhibited by the volume difference between a convex body and its floating body. In this talk I consider recent generalizations of the above notions obtained together with M.~Ludwig and E.~M.~Werner, where we consider the non-uniform case. This naturally gives rise to weighted floating bodies and a notion of weighted affine surface area for general convex bodies. In particular, from our results we can easily derive corresponding results for constant curvature spaces, i.e., the unit sphere or hyperbolic space. More recently, in joint work with C.~Sch{\"u}tt and E.~M.~Werner, we were able to give extensions also for the first term in the asymptotic expansion of the volume difference of a polytope and its weighted floating body, which now depends on weighted sum of the (complete) flags of the polytope. Our results raises the question, if again a similar behavior can be observed in the non-uniform random approximation of convex polytopes.

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