The radii of sections of origin-symmetric convex bodies and their applications

Abstract

Let V and W be any convex and origin-symmetric bodies in Rn . Assume that for some A ∈LRn→Rn, detA≠0, V is contained in the ellipsoid A−1B2n, where B2n is the unit Euclidean ball. We give a lower bound for the W-radius of sections of A−1V in terms of the spectral radius of A∗A and the expectations of ‖⋅‖V and ‖⋅‖Wo with respect to Haar measure on Sn−1⊂Rn. It is shown that the respective expectations are bounded as n→∞ in many important cases. As an application we offer a new method of evaluation of n-widths of multiplier operators. As an example we establish sharp orders of n-widths of multiplier operators Λ:LpMd→LqMd, 1<q≤2≤p<∞ on compact homogeneous Riemannian manifolds Md. Also, we apply these results to prove the existence of flat polynomials on Md.