On the Stability of the Volume Radius

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 √ n 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 Bárány 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 Rudin– Shapiro polynomials (see [3]). Later Kashin [9] used the same approach for finite dimensional analogues of Menshov’s correction theorem. Here we continue to study this property. We show that it holds not only for cubes but also for a wide class of bodies. It is then observed that the condition on the width of the bands which appeared in [3] is very close to Talagrand’s [10] description of bounded Gaussian processes. This observation permits us to restate the result in an invariant form which is more convenient for application. Moreover the result of [3] is extended to the intersection of a given body with a sequence of cylinders. This is made possible by a suitable generalization of the Khatri–Sidak theorem [11, 12]. Supported in part by the United States – Israel Binational Science Foundation.