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Journal of Convex Analysis 27 (2020), No. 4, 1303--1314
Copyright Heldermann Verlag 2020

On the Properties of Convex Functions over Open Sets

Yoel Drori
Google Research, Israel

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.

Keywords: Convex functions, open sets, descent lemma.

MSC: 26B25, 52A20, 52A41, 54C20.

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