"On the Auerbach-Mazur-Ulam problem.”

Thursday, September 28, 2017 - 16:00 to 16:50
427 Thackeray
Speaker Information
Michal Wojciechowski
Institute of Mathematics, Polish Academy of Sciences/Kent State

Abstract or Additional Information

In 1935 Auerbach, Mazur and Ulam proved that any centrally
symmetric body in R^3 with all two dimensional central sections affinely
equivalent to each other is an ellipsoid. This theorem was later generalized
to all odd dimensions by Gromov. The proofs are based on the algebraic
topology - nonexistence of a non-vanishing vector field tangent to the
sphere. This is the reason why in even dimensions the problem is still
open - this argument does not work there.

We present a new approach to the problem, that does not use the homological
properties of sphere. Under some mild smoothness condition we prove the
theorem in 3D using only differential properties of the body.
We hope that this approach will work in even dimensions.

Joint work with Bartłomiej Zawalski.