Generic differentiability of convex functions and monotone operators

[摘要] The aim of this paper is to investigate to what extent the known theory ofsubdifferentiability and generic differentiability of convex functions defined on opensets can be carried out in the context of convex functions defined on not necessarilyopen sets. Among the main results obtained I would like to mention a Kenderovtype theorem (the subdifferential at a generic point is contained in a sphere), ageneric Gâteaux differentiability result in Banach spaces of class S and a genericFréchet differentiability result in Asplund spaces. At least two methods can beused to prove these results: first, a direct one, and second, a more general one,based on the theory of monotone operators. Since this last theory was previouslydeveloped essentially for monotone operators defined on open sets, it was necessaryto extend it to the context of monotone operators defined on a larger class of sets,our "quasi open" sets. This is done in Chapter III. As a matter of fact, most ofthese results have an even more general nature and have roots in the theory ofminimal usco maps, as shown in Chapter II.

[发布机构] University:California Institute of Technology;Department:Physics, Mathematics and Astronomy

[关键词] Mathematics