Differentiability and local rotundity

Differentiability in Local Fields

Local Differentiability of Distance Functions

Recently Clarke, Stern and Wolenski characterized, in a Hilbert space, the closed subsets C for which the distance function dC is continuously differentiable everywhere on an open “tube” of uniform thickness around C. Here a corresponding local theory is developed for the property of dC being continuously differentiable outside of C on some neighborhood of a point x ∈ C. This is shown to be equ...

Strong Rotundity and Optimization

Control and Cybernetics Rotundity, Smoothness and Duality

The duality between smoothness and rotundity of functions is studied in a nonlinear abstract framework. Here smoothness is enlarged to subdifferentiability properties and rotundity is formulated by means of approximation properties.

Fréchet directional differentiability and Fréchet differentiability

Zaj́ıček has recently shown that for a lower semi-continuous real-valued function on an Asplund space, the set of points where the function is Fréchet subdifferentiable but not Fréchet differentiable is first category. We introduce another variant of Fréchet differentiability, called Fréchet directional differentiability, and show that for any realvalued function on a normed linear space, the se...