M-Estimates for Isotropic Convex Bodies and Their L q -Centroid Bodies

Abstract

Let K be a centrally-symmetric convex body in $$\mathbb{R}^{n}$$ and let $$\|\cdot \|$$ be its induced norm on $$\mathbb{R}^{n}$$ . We show that if K ⊇ rB 2 n then: $$\displaystyle{\sqrt{n}M(K)\leqslant C\sum _{k=1}^{n} \frac{1} {\sqrt{k}}\min \left (\frac{1} {r}, \frac{n} {k}\log \Big(e + \frac{n} {k}\Big) \frac{1} {v_{k}^{-}(K)}\right )}$$ where $$M(K) =\int _{S^{n-1}}\|x\|\,d\sigma (x)$$ is the mean-norm, C > 0 is a universal constant, and v 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 $$\mathbb{R}^{n}$$ and its L q -centroid bodies. In particular, we show that if K has isotropic constant L K then: $$\displaystyle{M(K)\leqslant \frac{C\log ^{2/5}(e + n)} {\root{10}\of{n}L_{K}}.}$$