On the fifth Busemann-Petty problem

Analysis Seminar
Wednesday, October 24, 2018 - 1:55pm for 1 hour (actually 50 minutes)
Skiles 005
Dmirty Ryabogin – Kent State University – ryabogin@math.kent.eduhttp://www.math.kent.edu/~ryabogin/
Galyna Livshyts
In 1956, Busemann and Petty posed a series of questions about symmetric convex bodies, of which only the first one has been solved.Their fifth problem asks the following.Let K be an origin symmetric convex body in the n-dimensional Euclidean space and let H_x be a hyperplane passing through the origin orthogonal to a unit direction x. Consider a hyperplane G parallel to H_x and supporting to K and let C(K,x)=vol(K\cap H_x)dist (0, G). (proportional to the volume of the cone spanned by the secion and the support point). If there exists a constant C such that for all directions x we have C(K,x)=C, does it follow that K is an ellipsoid?We give an affirmative answer to this problem for bodies sufficiently close to the Euclidean ball in the Banach Mazur distance.This is a joint work with Maria Alfonseca, Fedor Nazarov and Vlad Yaskin.