The distance to the boundary with respect to the Minkowski functional of a polytope

TL;DR

This paper investigates the regularity and explicit computation of the distance function to a domain boundary in relation to the Minkowski functional of a convex polytope, revealing new phenomena in specific cases.

Contribution

It provides new insights into the regularity properties and explicit formulas of the distance function with respect to the Minkowski functional of convex polytopes.

Findings

Regularity results for the distance function in certain cases

Explicit computations of the distance function in specific examples

Observation of new phenomena in the behavior of these distance functions

Abstract

We study the regularity of the distance function to the boundary of a domain in $\mathbb{R}^n$, with respect to the Minkowski functional of a convex polytope. We obtain the regularity of the distance function in certain cases. We also explicitly compute the distance function in a collection of examples and observe the new interesting phenomena that arise for such distance functions.