A Vertex-Skipping property for almost-minimizers of the relative perimeter in convex sets

TL;DR

This paper proves that almost-minimizers of the relative perimeter within a convex domain in three-dimensional space do not have boundary vertices inside the domain, revealing a geometric regularity property.

Contribution

It establishes a vertex-skipping property for almost-minimizers of the relative perimeter in convex sets, a novel geometric insight.

Findings

Closure of boundary does not contain vertices of the domain

Almost-minimizers exhibit a vertex-skipping property

Provides geometric regularity results for perimeter minimizers

Abstract

Given a convex domain $\Omega\subset \mathbb{R}^{3}$ and an almost-minimizer $E$ of the relative perimeter in $\Omega$, we prove that the closure of $\partial E \cap \Omega$ does not contain vertices of $\Omega$.