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Let $K$ be a centrally symmetric convex body in $\mathbb{R}^n$, and let $K^\circ$ be the polar or dual convex body. Define a statistic $e(K)$ as the expected value of $(\vec{x} \cdot \vec{y})^2$, where $\vec{x}$ is chosen randomly from $K$ and $\vec{y}$ is chosen randomly from $K^\circ$. Then for each fixed $n$, is $e(K)$ maximized when $K$ is an ellipsoid? The question is even open in two dimensions.
A much weaker conjecture is that the integral of $(\vec{x} \cdot \vec{y})^2$ over $K \times K^\circ$, as opposed to the average value, is maximized when $K$ is an ellipsoid. It is known that $K \times K^\circ$ has the most volume when $K$ is an ellipsoid; this fact is called Santaló's inequality.
It is known that the answer to the first conjecture is no if $K$ is not centrally symmetric, even if the origin is the only points fixed by the symmetries of $K$. (Central symmetry means specifically that $K = -K$.)