A sharp Santaló inequality in a slab with centrally symmetric sections

Abstract

We prove a sharp Santaló-type inequality for the product of $V(\phi )$ and $V$ of the Legendre transform of $\phi$, where $\phi$ is an even geometric convex function on ${\mathbb{R}}^n$ and $V(\phi )=\int (1+\phi {)}^{-(n+1)}$. For this size functional, the Legendre transform and functional polarity give the same volume, so the same inequality holds for both dualities. Through a projective correspondence, the problem is equivalent to a new Santaló-type inequality with respect to standard volume and standard polarity, on a class of not necessarily centered convex bodies. This class consists of convex bodies $K\subset {\mathbb{R}}^n\times [-1,1]$ containing $\{0\}\times [-1,1]$, for which every horizontal section is centrally symmetric about the distinguished vertical axis. We determine the sharp upper bound for ${\mathrm{vol}}_{n+1}(K){\mathrm{vol}}_{n+1}(K^{\circ })$ in this class and classify all equality cases. When $n=1$, the extremizers are the disk and its horizontal linear images. When $n\ge 2$, the ball is no longer a maximizer, and the extremizer is unique up to horizontal linear transformations and the reflection $t\mapsto -t$. The symmetric Santaló inequality applied to horizontal sections, together with monotone transport, reduces the problem to a maximization problem for increasing curves in the square $[-1,1{]}^2$. We solve this problem using a global potential when $n=1$ and two Hamilton-Jacobi branches when $n\ge 2$. Finally, in dimension one we provide a sharp family of functional Santaló inequalities for the Legendre duality, interpolating between the volume $V$ and the exponential volume $\int \exp (-\phi )$, with centered quadratics as the only equality cases.

48 pages, 6 figures