# On the convexity of piecewise-defined functions

arXiv: https://arxiv.org/abs/1408.3771 · HTML: https://mlsanity.com/paper/1408.3771/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 16 Aug 2014.

## Built on by (most cited first, 2 of 2)

- [Convexity in ReLU Neural Networks: beyond ICNNs?](https://mlsanity.com/paper/2501.03017/) (2025)
- [Spurious Stationarity and Hardness Results for Bregman Proximal-Type Algorithms](https://mlsanity.com/paper/2404.08073/) (2024)

## Abstract (arXiv)

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.