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

摘要

Let $K \subset {\mathbb R}^2$ be an $o$-symmetric convex body, and $K^*$ itspolar body. Then we have $|K|\cdot |K^*| \ge 8$, with equality if and only if$K$ is a parallelogram. ($| \cdot |$ denotes volume). If $K \subset {\mathbbR}^2$ is a convex body, with $o \in {\text{int}}\,K$, then $|K|\cdot |K^*| \ge27/4$, 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. Thesetheorems are due to Mahler and Reisner, Mahler and Meyer, and to Eggleston,respectively. We show an analogous theorem: if $K$ has $n$-fold rotationalsymmetry about $o$, then $|K|\cdot |K^*| \ge n^2 \sin ^2 ( \pi /n)$, withequality if and only if $K$ is a regular $n$-gon of centre $o$. We will alsogive stability variants of these four inequalities, both for the body, and forthe centre of polarity. For this we use the Banach-Mazur distance (fromparallelograms, or triangles), or its analogue with similar copies rather thanaffine transforms (from regular $n$-gons), respectively. The stability variantsare sharp, up to constant factors. We extend the inequality $|K|\cdot |K^*| \gen^2 \sin ^2 ( \pi /n)$ 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 isa stability estimate for the area product of two sectors of convex bodies polarto each other. To several of our statements we give several proofs; inparticular, we give a new proof for the theorem of Mahler-Reisner.