$\Psi_2$-Estimates for Linear Functionals on Zonoids

Abstract

Let K be a convex body in ${\mathbb R}^n$ with centre of mass at the origin and volume |K| = 1. We prove that if $K\subseteq\alpha\sqrt{n}B_2^n$ where B₂ⁿ is the Euclidean unit ball, then there exists $\theta\in S^{n-1}$ such that

$$\|\langle \cdot ,\theta\rangle \|_{L_{\psi_{2}}(K)}\leq c\alpha \|\langle \cdot ,\theta\rangle\|_{L_1(K)}, \qquad (*)$$

where c > 0 is an absolute constant. In other words, “every body with small diameter has $\psi_2$-directions”. This criterion applies to the class of zonoids. In the opposite direction, we show that if an isotropic convex body K of volume 1 satisfies (*) for every direction $\theta\in S^{n-1}$, then $K\subseteq C\alpha^2\sqrt{n}\log nB_2^n$, where C > 0 is an absolute constant.