Recently Clarke, Stern and Wolenski characterized, in a Hilbert space, the closed subsets C for which the distance function d C is continuously differentiable everywhere on an open " tube " of uniform thickness around C. Here a corresponding local theory is developed for the property of d C being continuously differentiable outside of C on some neighborhood… (More)
RÉSUMÉ. Nous démontrons la formule du sous-différentiel de la somme de deux fonctions convexes sous une condition de qualification très générale et nous prouvons que cette condition est impliquée par toutes les conditions de qualification utilisées jusqu'ici. ABSTRACT. The subdifferential formula for the sum of two convex functions defined on a locally… (More)
In this work we introduce for extended real valued functions, defined on a Banach space X, the concept of K directionally Lipschitzian behavior, where K is a bounded subset of X. For different types of sets K (e.g., zero, singleton, or compact), the K directionally Lipschitzian behavior recovers well-known concepts in variational analysis (locally… (More)
We prove in the general setting of lower semicontinuous functions on Banach spaces the relation between the Rockafellar directional derivative and the mixed lower limit of the lower Dini derivatives. As a byproduct we derive the famous inclusions of tangent cones of closed sets in Banach spaces. The results are established using as principal tool… (More)
We prove that a lower semicontinuous function defined on a reflexive Banach space is convex if and only if its Clarke subdifferential is monotone.