# Rigidity results for variational infinity ground states

## TL;DR

This paper establishes two rigidity theorems for variational infinity ground states in convex domains, showing they match boundary distance functions under specific conditions, leading to geometric characterizations of the domain.

## Contribution

It introduces new rigidity results linking the properties of ground states to the geometry of convex domains, specifically characterizing stadium-like shapes.

## Key findings

- Ground states coincide with boundary distance functions under certain conditions.
- Convex domains with specific ground state properties are characterized as stadium-like.
- Rigidity results connect PDE solutions to geometric domain features.

## Abstract

We prove two rigidity results for a variational infinity ground state $u$ of an open bounded convex domain $\Omega \subset \mathbb{R}^n$. They state that $u$ coincides with a multiple of the distance from the boundary of $\Omega$ if either $|\nabla u|$ is constant on $\partial \Omega$, or $u$ is of class $C ^ {1,1}$ outside the high ridge of $\Omega$. Consequently, in both cases $\Omega$ can be geometrically characterized as a "stadium-like domain".