It is proved that the simplex is a strict local minimum for the volume product, P(K)=min z∈int(K)|K||Kz|, in the Banach-Mazur space of n-dimensional (classes of) convex bodies. Linear local stability in the neighborhood of the simplex is proved as well. The proof consists of an extension to the non-symmetric setting of methods that were recently introduced by Nazarov, Petrov, Ryabogin and Zvavitch, as well as proving results of independent interest concerning stability of square order of volumes of polars of non-symmetric convex bodies.
|Number of pages||14|
|State||Published - Jan 2011|
Bibliographical noteFunding Information:
Acknowledgements. It is with great pleasure that we thank Karoly Böröczky, Mathieu Meyer, Dmitry Ryabogin and Artem Zvavitch for very helpful advice during the preparation of this paper. The first author was supported in part by U.S. National Science Foundation grant DMS-0652684. The second author was supported in part by the France–Israel Research Network Program in Mathematics contract #3-4301.
ASJC Scopus subject areas
- Mathematics (all)