On the convexity of piecewise-defined functions

TL;DR

This paper investigates conditions under which a piecewise-defined function is globally convex, providing verifiable criteria and examples to determine convexity or its absence in such functions.

Contribution

It offers new, practical conditions to verify the convexity of piecewise-defined functions, enhancing understanding of their global convexity properties.

Findings

Provided verifiable conditions for convexity of piecewise functions

Illustrated results with multiple examples

Identified scenarios where convexity does not hold

Abstract

Functions that are piecewise defined are a common sight in mathematics while convexity is a property especially desired in optimization. Suppose now a piecewise-defined function is convex on each of its defining components - when can we conclude that the entire function is convex? In this paper we provide several convenient, verifiable conditions guaranteeing convexity (or the lack thereof). Several examples are presented to illustrate our results.