Strengthened volume inequalities for $L_p$ zonoids of even isotropic measures

Abstract

We strengthen the volume inequalities for $L_p$ zonoids of even isotropic measures and for their duals, which are originally due to Ball, Barthe, and Lutwak, Yang, and Zhang. The special case $p=\infty$ yields a stability version of the reverse isoperimetric inequality for centrally symmetric convex bodies. Adding to known inequalities and stability results for the reverse isoperimetric inequality of arbitrary convex bodies, we state a conjecture on volume inequalities for $L_p$ zonoids of general centered (non-symmetric) isotropic measures. We achieve our main results by strengthening Barthe’s measure transportation proofs of the rank one case of the geometric Brascamp–Lieb and reverse Brascamp–Lieb inequalities by estimating the derivatives of the transportation maps for a special class of probability density functions. Based on our argument, we phrase a conjecture about a possible stability version of the Brascamp–Lieb and reverse Brascamp–Lieb inequalities. We also establish some geometric properties of the distribution of general isotropic measures that are essential to our argument. In particular, we prove a measure theoretic analog of the Dvoretzky–Rogers lemma.