Sets resilient to erosion

TL;DR

This paper characterizes sets in Euclidean space that remain similar to their eroded versions, revealing geometric constraints for convex sets and a link to scale-invariant fractal sets for nonconvex ones.

Contribution

It provides a surprising characterization of resilient sets, connecting convex geometric constraints with nonconvex fractal-like structures.

Findings

Convex resilient sets satisfy specific geometric constraints.

Nonconvex resilient sets correspond to scale-invariant fractal sets.

Resilience to erosion relates to set self-similarity and invariance.

Abstract

The erosion of a set in Euclidean space by a radius r>0 is the subset of X consisting of points at distance >/-r from the complement of X. A set is resilient to erosion if it is similar to its erosion by some positive radius. We give a somewhat surprising characterization of resilient sets, consisting in one part of simple geometric constraints on convex resilient sets, and, in another, a correspondence between nonconvex resilient sets and scale-invariant (e.g., 'exact fractal') sets.