Remarks on planar Blaschke-Santaló inequality

Abstract

We prove the Blaschke-Santaló inequality restricted to $n$-gons: the extremal polygons are the affine regular $n$-gons. If either the John or the Löwner ellipse of a planar $o$-symmetric convex body $K$ is the unit circle about $o$, then a sharpening of the Blaschke-Santaló inequality holds: even the aritmetic mean $\left(V(K)+V(K^{\ast })\right)/2$ is at least $\pi$. We give stability variants of the Blaschke-Santaló inequality for the plane. If for some $n\ge 3$ the planar convex body $K$ is $n$-fold rotationally symmetric about $o$, then we give the exact maximum of $V(K^{\ast })$, as a function of $V(K)$ and the area of either the John or the Löwner ellipse.

21 pages

References in corpus (1)

• Stability of some versions of the Prékopa-Leindler inequality

Cited by in corpus (1)

• Inverse Blaschke-Santaló inequality for convex curves enclosing the origin several times