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