On the reverse isodiametric problem and Dvoretzky-Rogers-type volume bounds

The isodiametric inequality states that the Euclidean ball maximizes the volume among all convex bodies of a given diameter. We are motivated by a conjecture of Makai Jr. on the reverse question: Every convex body has a linear image whose isodiametric quotient is at least as large as that of a regular simplex. We relate this reverse isodiametric problem to minimal volume enclosing ellipsoids and to the Dvoretzky-Rogers-type problem of finding large volume simplices in any decomposition of the identity matrix. As a result, we solve the reverse isodiametric problem for o-symmetric convex bodies and obtain a strong asymptotic bound in the general case. Using the Cauchy-Binet formula for minors of a product of matrices, we obtain Dvoretzky-Rogers-type volume bounds which are of independent interest.


Year:
Apr 16 2018
Dataset(s):
url: https://arxiv.org/abs/1804.05009
Laboratories:




 Record created 2018-05-16, last modified 2018-10-07


Rate this document:

Rate this document:
1
2
3
 
(Not yet reviewed)