There are Thin Minimizers of the L1TV Functional

TL;DR

This paper reveals that L1TV minimizers can have a global reach smaller than the local lower bound, challenging assumptions about their boundary properties through specific examples.

Contribution

It demonstrates that the global reach of L1TV minimizers can be smaller than the local lower bound, providing new insights into their geometric properties.

Findings

Local boundary reach is bounded below by 1/lambda.

Global reach can be smaller than the local bound.

Examples of minimizers not equal to union of 1/lambda-balls.

Abstract

In this paper we show the surprising results that while the local reach of the boundary of an L1TV minimizer is bounded below by 1/lambda, the global reach can be smaller. We do this by demonstrating several example minimizing sets not equal to the union of the 1/lambda -balls they contain.