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A Santalo-type Inequality for the J Transform Florentin, Dan


n recent years, it was proven that there exist precisely four order isomorphisms acting in the class of geometric convex functions. These are the Legendre transform ${\cal L}$, the geometric duality transform ${\cal A}$, their composition ${\cal J}$, and the identity. It is known that ${\cal L}$ and ${\cal A}$ satisfy Santal\'{o}-type inequalities, e.g. the quantity $M(f) = {\rm Vol}(f)\cdot{\rm Vol}({\cal L}f)$ is bounded from above and below (here ${\rm Vol}(f)$ stands for the integral over ${\mathbb R}^n$ of $e^{-f}$). We prove similar (asymptotically sharp) bounds for the quantity $M^{\cal J} (f) = {\rm Vol}( {\cal J} f) / {\rm Vol}(f)$, and describe the extremal functions.

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