M-estimates for isotropic convex bodies and their Lq-centroid bodies

Abstract

Let K be a centrally-symmetric convex body in Rn and let ||· || be its induced norm on Rn. We show that if K 2 ⊇ Bn2 then: (Formula presented) where M(K) = ∫Sn-1 ||x|| da(x) is the mean-norm, C> 0 is a universal constant, andV-k (K) denotes the minimal volume-radius of a k-dimensional orthogonal projection of K. We apply this result to the study of the mean-norm of an isotropic convex body K in Rn and its Lq-centroid bodies. In particular, we show that if K has isotropic constant LK then: (Formula presented).