Remarks on planar Blaschke-Santal\'o inequality

We prove the Blaschke-Santal\'o inequality restricted to $n$-gons: the extremal polygons are the affine regular $n$-gons. If either the John or the L\"owner ellipse of a planar $o$-symmetric convex body $K$ is the unit circle about $o$, then a sharpening of the Blaschke-Santal\'o inequality holds: even the aritmetic mean $\left( V(K) + V( K^*) \right) /2 $ is at least $\pi $. We give stability variants of the Blaschke-Santal\'o 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 $V(K^*)$, as a function of $V(K)$ and the area of either the John or the L\"owner ellipse.