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Mathematics Colloquium

Dmitri Ryabogin
Kent State University

Title: A solution to a local version of the fifth Busemann-Petty Problem.
Date: Friday, September 27, 2019
Place and Time: Room 101, Love Building, 3:35-4:25 pm
Refreshments: Room 204, Love Building, 3:00 pm

Abstract. 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). 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.