Analysis Seminar: Dmitry Ryabogin (Kent) "On a local version of the fifth Busemann-Petty Problem"

Title: On a local version of the fifth Busemann-Petty Problem
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.


Wed, 19/12/2018 - 12:00 to 13:00


Ross Building, Room 70