On Mahler's conjecture for even s-concave functions in dimensions 1 and 2

In this paper, we establish different sharp forms of Mahler’s conjecture for $s$-concave even functions in dimensions $n$, for $n=1$ and $2$, for $s>-1\mathrm{/}n$, thus generalizing our previous results in \cite{FN} on log-concave even functions in dimension 2, which corresponds to the case $s=0$. The functional volume product of an even $s$-concave function $g$ is $${\int }_{{\mathbb{R}}^n}g(x)dx{\int }_{{\mathbb{R}}^n}{\mathcal{L}}_{s}g(y)dy,$$ where ${\mathcal{L}}_{s}g$ is the $s$-polar function associated to $g$. The analogue of Mahler’s conjecture for even $s$-concave functions postulates that this quantity is minimized for the indicatrix of a cube for any $s>-1\mathrm{/}n$. In dimension $n=1$, we prove this conjecture for all $s\in (-1,0)$ (the case $s\ge 0$ was established by the first author and Mathieu Meyer in \cite[page 17]{FM10}). In dimension $n=2$, we only consider the case $1\mathrm{/}s\in \mathbb{Z}$: for $s>0$, we establish Mahler’s conjecture for general $s$-concave even functions; for $s<0$, the situation is more involved, we only prove a sharp inequality for $s$-concave functions $g$ such that $g^s$ admits an asymptote in every direction. Notice that this set of functions is quite natural to consider, when $s<0$, since it is the largest subset of $s$-concave functions stable by $s$-duality.