The logarithmic Minkowski inequality for cylinders

TL;DR

This paper establishes a logarithmic Minkowski inequality for o-symmetric cylinders and convex bodies in three-dimensional space, identifying conditions for equality related to cylinders.

Contribution

It proves a new logarithmic Minkowski inequality specifically for o-symmetric cylinders and convex bodies in three dimensions, with a characterization of equality cases.

Findings

Inequality holds for o-symmetric cylinders and convex bodies in R^3.

Equality occurs if and only if the bodies are relative cylinders.

The result extends Minkowski-type inequalities to a new class of geometric objects.

Abstract

In this paper, we prove that if $K$ is an $o$-symmetric cylinder and $L$ is an $o$-symmetric convex body in $\mathbb R^3$, then the logarithmic Minkowski inequality \[\frac{1}{V(K)}\int_{\mathbb S^{2}}\log\frac{h_L}{h_K}\,dV_K\geq\frac{1}{3}\log\frac{V(L)}{V(K)}\] holds, with equality if and only if $K$ and $L$ are relative cylinders.