Geodesic
On the Properties of Convex Functions over Open Sets
Journal of convex analysis, Tome 27 (2020) no. 4, pp. 1303-1314
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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.
Classification : 26B25, 52A20, 52A41, 54C20
Mots-clés : Convex functions, open sets, descent lemma 
@article{JCA_2020_27_4_JCA_2020_27_4_a12,
     author = {Y. Drori},
     title = {On the {Properties} of {Convex} {Functions} over {Open} {Sets}},
     journal = {Journal of convex analysis},
     pages = {1303--1314},
     year = {2020},
     volume = {27},
     number = {4},
     url = {http://geodesic.mathdoc.fr/item/JCA_2020_27_4_JCA_2020_27_4_a12/}
}
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Y. Drori. On the Properties of Convex Functions over Open Sets. Journal of convex analysis, Tome 27 (2020) no. 4, pp. 1303-1314. http://geodesic.mathdoc.fr/item/JCA_2020_27_4_JCA_2020_27_4_a12/