Nonconvex Lipschitz function in plane which is locally convex outside a discontinuum

TL;DR

This paper constructs a Lipschitz function in the plane that is locally convex outside a totally disconnected set but not globally convex, disproving a previously claimed theorem.

Contribution

It provides a counterexample to a theorem about convexity properties of Lipschitz functions in the plane.

Findings

Constructed a Lipschitz function that is locally convex outside a totally disconnected set

Disproved a previously claimed theorem on convexity of Lipschitz functions

Showed that local convexity outside a discontinuum does not imply global convexity

Abstract

We construct a Lipschitz function on $\er^{2}$ which is locally convex on the complement of some totally disconnected compact set but not convex. Existence of such function disproves a theorem that appeared in a paper by L. Pasqualini and was also cited by other authors.