A bornological approach to rotundity and smoothness applied to approximation.
A criterion of Γ-nullness and differentiability of convex and quasiconvex functions
We introduce a criterion for a set to be Γ-null. Using it we give a shorter proof of the result that the set of points where a continuous convex function on a separable Asplund space is not Fréchet differentiable is Γ-null. Our criterion also implies a new result about Gâteaux (and Hadamard) differentiability of quasiconvex functions.
A generalized differential of real functionals.
A Lipschitz function which is on a.e. line need not be generically differentiable
We construct a Lipschitz function f on X = ℝ ² such that, for each 0 ≠ v ∈ X, the function f is smooth on a.e. line parallel to v and f is Gâteaux non-differentiable at all points of X except a first category set. Consequently, the same holds if X (with dimX > 1) is an arbitrary Banach space and “a.e.” has any usual “measure sense”. This example gives an answer to a natural question concerning the author’s recent study of linearly essentially smooth functions (which generalize essentially smooth...
A new subdifferential in quasiconvex analysis.
A note on the characterization of the global maxima of a (tangentially) convex function over a convex set.
A one-sided version of Alexiewicz-Orlicz's differentiability theorem.
Modificando adecuadamente el método de un trabajo olvidado [1], probamos que si una aplicación continua, de un subconjunto abierto no vacío U de un espacio vectorial topológico metrizable separable y de Baire E, en un espacio localmente convexo, es direccionalmente diferenciable por la derecha en U según un subconjunto comagro de E, entonces, es genéricamente Gâteaux diferenciable en U. Nuestro resultado implica que cualquier espacio vectorial topológico, metrizable, separable y de Baire, es débilmente...
A polynomial of degree four not satisfying Rolle’s Theorem in the unit ball of
We give an example of a fourth degree polynomial which does not satisfy Rolle’s Theorem in the unit ball of .
A remark on localized weak precompactness in Banach spaces
We give a characterization of -weakly precompact sets in terms of uniform Gateaux differentiability of certain continuous convex functions.
A result on equiabsolute integrability
We prove the equiabsolute integrability of a class of gradients, for functions in . The present result appears as the localized version of well-known classical theorems.
A simple proof of the Rademacher theorem