More on logarithmic sums of convex bodies

Abstract

We prove that the log-Brunn-Minkowski inequality (log-BMI) for the Lebesque measure in dimension $n$ would imply the log-BMI and, therefore, the B-conjecture for any log-concave density in dimension $n$. As a consequence, we prove the log-BMI and the B-conjecture for any log-concave density, in the plane. Moreover, we prove that the log-BMI reduces to the following: For each dimension $n$, there is a density $f_n$, which satisfies an integrability assumption, so that the log-BMI holds for parallelepipeds with parallel facets, for the density $f_n$. As byproduct of our methods, we study possible log-concavity of the function $t\mapsto |(K{+}_p\cdot e^{t}L{)}^{\circ }|$, where $p\ge 1$ and $K$, $L$ are symmetric convex bodies, which we are able to prove in some instances and as a further application, we confirm the variance conjecture in a special class of convex bodies. Finally, we establish a non-trivial dual form of the log-BMI.

Minor corrections, some additional references, agnowledgement

References in corpus (3)

• The Logarithmic Minkowski Problem
• Thin shell implies spectral gap up to polylog via a stochastic localization scheme
• The Dual Orlicz-Brunn-Minkowski Theory

Cited by in corpus (3)

• Gaussian mixtures: entropy and geometric inequalities
• $L^p$-Minkowski Problem under Curvature Pinching
• Isotropic Measures and Maximizing Ellipsoids: Between John and Loewner