Sharp L1 Inequalities for Sup-Convolution

Given a compact convex domain $C\subset \mathbb{R}^k$ and bounded measurable functions $f_1,\ldots,f_n:C\to \mathbb{R}$, define the sup-convolution $(f_1\ast \ldots \ast f_n)(z)$ to be the supremum average value of $f_1(x_1),\ldots,f_n(x_n)$ over all $x_1,\ldots,x_n\in C$ which average to $z$. Continuing the study by Figalli and Jerison and the present authors of linear stability for the Brunn-Minkowski inequality with equal sets, for $k\le 3$ we find the optimal constants $c_{k,n}$ such that $$\int_C f^{\ast n}(x)-f(x) dx \ge c_{k,n}\int_C\text{co}(f)(x)-f(x) dx$$ where $\text{co}(f)$ is the upper convex hull of $f$. Additionally, we show $c_{k,n}=1-O(\frac{1}{n})$ for fixed $k$ and prove an analogous optimal inequality for two distinct functions. The key geometric insight is a decomposition of polytopal approximations of $C$ into hypersimplices according to the geometry of the set of points where $\text{co}(f)$ is close to $f$.