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

On the Properties of Convex Functions over Open Sets