# On a local version of the fifth Busemann-Petty Problem

Seminar
Speaker
Prof. D. Ryabogin, Kent State University, Ohio, USA
Date
17/12/2018 - 15:30 - 14:00Add to Calendar 2018-12-17 14:00:00 2018-12-17 15:30:00 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. 2nd floor Colloquium Room, Building 216 אוניברסיטת בר-אילן - המחלקה למתמטיקה mathoffice@math.biu.ac.il Asia/Jerusalem public
Place
2nd floor Colloquium Room, Building 216
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.

תאריך עדכון אחרון : 17/12/2018