On Removable Sets For Convex Functions

TL;DR

This paper establishes conditions under which certain closed sets in Euclidean space are removable for convex functions, showing that some large sets are not removable and providing counterexamples to a related conjecture.

Contribution

It provides a sufficient condition for c-removability of sets and constructs counterexamples to a conjecture about the smallness of sets ensuring convex extension.

Findings

No generalized rectangle of positive measure in R^2 is c-removable.

A set with the property of unique convex extension need not be intervally thin.

The paper answers a question about the relationship between set smallness and convex extension negatively.

Abstract

In the present article we provide a sufficient condition for a closed set F in R^d to have the following property which we call c-removability: Whenever a function f:R^d->R is locally convex on the complement of F, it is convex on the whole R^d. We also prove that no generalized rectangle of positive Lebesgue measure in R^2 is c-removable. Our results also answer the following question asked in an article by Jacek Tabor and Jozef Tabor [J. Math. Anal. Appl. 365 (2010)]: Assume the closed set F in R^d is such that any locally convex function defined on R^d\F has a unique convex extension on R^d. Is F necessarily intervally thin (a notion of smallness of sets defined by their "essential transparency" in every direction)? We prove the answer is negative by finding a counterexample in R^2.