# On the Properties of Convex Functions over Open Sets

arXiv: https://arxiv.org/abs/1812.02419 · HTML: https://mlsanity.com/paper/1812.02419/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 6 Dec 2018.

## Built on by (most cited first, 3 of 3)

- [Glocal Smoothness: Line search and adaptive step sizes can help in theory too!](https://mlsanity.com/paper/2506.12648/) (2025)
- [Small Gradient Norm Regret for Online Convex Optimization](https://mlsanity.com/paper/2601.13519/) (2026)
- [ProofLoom: Proof-Obligation-Driven Theory Construction for Autoformalizing Research-Level Stochastic Optimization](https://mlsanity.com/paper/2609.34960/) (2026)

## Abstract (arXiv)

We consider the class of smooth convex functions defined over an open convex set. We show that this class is essentially different than the class of smooth convex functions defined over the entire linear space by exhibiting a function that belongs to the former class but cannot be extended to the entire linear space while keeping its properties. We proceed by deriving new properties of the class under consideration, including an inequality that is strictly stronger than the classical Descent Lemma.