A strengthening of the Blaschke-Santal\'o inequality for $o$-symmetric planar convex bodies

Abstract

We verify the inequality $$ \frac{|K|}{|E|}+\frac{|K^*|}{|E^*|}\leq 2 $$ for any $o$-symmetric convex body $K\subset\mathbb{R}^2$ where $E$ is either the John ellipse of maximal area contained in $K$ or the minimal area L\"owner ellipse containing $K$. The analogous estimate may not hold if $K$ is a planar but the assumption of $o$-symmetry is dropped, or if $K$ is $o$-symmetric convex body in $\mathbb{R}^n$ for $n\geq 3$. Our new inequality strengthens the Blaschke-Santal\'o inequality for $o$-symmetric convex bodies $K\subset\mathbb{R}^2$ with an error term of optimal order.