A sharp Santal\'o inequality in a slab with centrally symmetric sections

Abstract

We prove a sharp Santal\'o-type inequality for the product of $V(\varphi)$ and $V$ of the Legendre transform of $\varphi$, where $\varphi$ is an even geometric convex function on $\mathbb{R}^n$ and $V(\varphi)=\int(1+\varphi)^{-(n+1)}$. 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 Santal\'o-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 $\operatorname{vol}_{n+1}(K)\operatorname{vol}_{n+1}(K^\circ)$ 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 Santal\'o 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 Santal\'o inequalities for the Legendre duality, interpolating between the volume $V$ and the exponential volume $\int \exp(-\varphi)$, with centered quadratics as the only equality cases.