On an extension of the Blaschke-Santalo inequality

TL;DR

This paper investigates an extension of the Blaschke-Santalo inequality by analyzing a specific functional involving a convex body and its polar, confirming the conjecture for p-balls.

Contribution

The authors verify the conjecture that the functional is maximized by the Euclidean ball when restricted to p-balls, extending the inequality's understanding.

Findings

Confirmed the conjecture for p-balls

Extended the Blaschke-Santalo inequality to a new functional

Provided new insights into convex body polar relationships

Abstract

Let $K$ be a convex body and $K^\circ$ its polar body. Call $\phi(K)=\frac{1}{|K||K^\circ|}\int_K\int_{K^\circ}< x,y>^2 dxdy$. It is conjectured that $\phi(K)$ is maximum when $K$ is the euclidean ball. In particular this statement implies the Blaschke-Santalo inequality. We verify this conjecture when $K$ is restricted to be a $p$--ball.