Volume product of planar polar convex bodies --- lower estimates with stability

TL;DR

This paper investigates volume product inequalities for planar convex bodies with symmetry, extending classical results with stability estimates and characterizations of equality cases, including new proofs and generalizations.

Contribution

It introduces stability variants of classical volume product inequalities for symmetric and general convex bodies, with sharp bounds and new proofs, extending to bodies with rotational symmetry.

Findings

Established stability estimates for volume product inequalities.

Extended inequalities to bodies with n-fold rotational symmetry.

Provided new proofs for classical theorems.

Abstract

Let $K \subset {\mathbb R}^2$ be an $o$-symmetric convex body, and $K^*$ its polar body. Then we have $|K|\cdot |K^*| \ge 8$, with equality if and only if $K$ is a parallelogram. ($| \cdot |$ denotes volume). If $K \subset {\mathbb R}^2$ is a convex body, with $o \in {\text{int}}\,K$, then $|K|\cdot |K^*| \ge 27/4$, with equality if and only if $K$ is a triangle and $o$ is its centroid. If $K \subset {\mathbb R}^2$ is a convex body, then we have $|K| \cdot |[(K-K)/2)]^* | \ge 6$, with equality if and only if $K$ is a triangle. These theorems are due to Mahler and Reisner, Mahler and Meyer, and to Eggleston, respectively. We show an analogous theorem: if $K$ has $n$-fold rotational symmetry about $o$, then $|K|\cdot |K^*| \ge n^2 \sin ^2 ( \pi /n)$, with equality if and only if $K$ is a regular $n$-gon of centre $o$. We will also give stability variants of these four inequalities, both for the body, and for the centre of polarity. For this we use the Banach-Mazur distance (from parallelograms, or triangles), or its analogue with similar copies rather than affine transforms (from regular $n$-gons), respectively. The stability variants are sharp, up to constant factors. We extend the inequality $|K|\cdot |K^*| \ge n^2 \sin ^2 ( \pi /n)$ to bodies with $o \in {\text{int}}\,K$, which contain, and are contained in, two regular $n$-gons, the vertices of the contained $n$-gon being incident to the sides of the containing $n$-gon. Our key lemma is a stability estimate for the area product of two sectors of convex bodies polar to each other. To several of our statements we give several proofs; in particular, we give a new proof for the theorem of Mahler-Reisner.