Sharp L1 Inequalities for Sup-Convolution

TL;DR

This paper establishes sharp inequalities relating sup-convolution and convex hulls of functions on convex domains, with optimal constants for dimensions up to three, advancing the understanding of stability in geometric inequalities.

Contribution

The authors derive the optimal constants in L1 inequalities for sup-convolution versus convex hulls on convex domains for dimensions up to three, extending stability results.

Findings

Optimal constants c_{k,n} found for k ≤ 3

Proved c_{k,n} approaches 1 as n increases

Established analogous inequalities for pairs of functions

Abstract

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 $f$.