A strengthening of the Blaschke-Santaló inequality for $o$-symmetric planar convex bodies

We verify the inequality $$\frac{\mathrm{\mid }K\mathrm{\mid }}{\mathrm{\mid }E\mathrm{\mid }}+\frac{\mathrm{\mid }{K}^{\ast }\mathrm{\mid }}{\mathrm{\mid }{E}^{\ast }\mathrm{\mid }}\le 2$$ 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 Löwner 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 $n\ge 3$. Our new inequality strengthens the Blaschke-Santaló inequality for $o$-symmetric convex bodies $K\subset {\mathbb{R}}^2$ with an error term of optimal order.