On the Stability of the Volume Radius

Summary

The volume radius of a given n-dimensional body is the radius of a euclidean ball having the same volume as this body. We prove that the volume radius of a given convex symmetric n-dimensional body with diameter at most is almost equal to the volume radius of a body obtained by the intersection of this body with n other bodies whose polars are bounded by 1 mean width.

In the last decade, interest in the problem of bounds for volumes of convex bodies was renewed mainly because of its applications to Banach Space Geometry and related topics. At the end of the 80's sharp bounds for volume radius of convex polytopes with given distance between antipodal faces were found independently by several authors: Carl and Pajor [1], Bourgain, Lindenstrauss and Milman [2], Gluskin [3]. Closely related results were obtained by Vaaler [4], Dilworth and Szarek [5] and Baniny and Furedi [6]. See also Ball and Pajor [7] where, following Kashin's conjecture, the problem was considered as a limiting case of a series of Vaaler-type results. Moreover in [3] it was observed that the volume radius of a unit cube has a certain stability property with respect to cutting the cube by a sequence of bands (see Proposition 1 below for the exact formulation). Some of Kashin's ideas enabled us to use this property for an alternative proof of Spencer's theorem [8] on a lacunary analogue of the RudinShapiro polynomials (see [3]). Later Kashin [9] used the same approach for finite dimensional analogues of Menshov's correction theorem.