Department of Mathematics and Statistics at the Faculty of Science
University of Calgary
On the extremal distance between two convex bodies
Submitted by ppaterso on Mon, 03/23/2009 - 10:33am.
Mar 27 2009 - 4:00pm
Mar 27 2009 - 5:00pm
Speaker:
Marton Naszodi (Dept. of Math. and Stats., University of Alberta)
Location:
MS 431Discrete Geometry Seminar
We consider the following modified version of the Banach--Mazur distance
of convex bodies in $\Re^n$: $d(K,L)=\inf\{|\lambda| : \lambda\in \Re, \tilde{K}\subset
\tilde{L}\subset \lambda\tilde{K}\}$ , where the infimum is taken over all non-degenerate
affine images $\tilde{K}$ and $\tilde{L}$ of $K$ and $L$. In [GLMP], it is shown that
for any two convex bodies $d(K,L)\leq n$, moreover, if $K$ is a simplex and $L=-L$ then $d(K,L)=n$. It is natural to ask whether equality is only attained if one of the sets is a simplex. Leichtweiss, and later Palmon proved that if $d(K,B_2^n)=n$, where $B_2^n$ is the Euclidean ball, then $K$ is the simplex. We prove the affirmative answer to the question in the case when $L$ is strictly convex or smooth, thus obtaining a strong generalization of the result of Leichtweiss and Palmon. This is a joint work with Carlos Jimenez (University of Seville).
[GLMP] Y. Gordon, A. E. Litvak, M. Meyer, and A. Pajor: John’s decomposition in the general case and applications, J. Differential Geom. 68 (2004), no. 1, 99–119.
for any two convex bodies $d(K,L)\leq n$, moreover, if $K$ is a simplex and $L=-L$ then $d(K,L)=n$. It is natural to ask whether equality is only attained if one of the sets is a simplex. Leichtweiss, and later Palmon proved that if $d(K,B_2^n)=n$, where $B_2^n$ is the Euclidean ball, then $K$ is the simplex. We prove the affirmative answer to the question in the case when $L$ is strictly convex or smooth, thus obtaining a strong generalization of the result of Leichtweiss and Palmon. This is a joint work with Carlos Jimenez (University of Seville).
[GLMP] Y. Gordon, A. E. Litvak, M. Meyer, and A. Pajor: John’s decomposition in the general case and applications, J. Differential Geom. 68 (2004), no. 1, 99–119.