Local differentiability of distance functions

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...

Differentiability of Distance Functions and a Proximinal Property Inducing Convexity

In a normed linear space X, consider a nonempty closed set K which has the property that for some r > 0 there exists a set of points xo € X\K, d(xoK) > r, which have closest points p(xo) € K and where the set of points xo — r((xo — p(xo))/\\xo — p(zo)||) is dense in X\K. If the norm has sufficiently strong differentiability properties, then the distance function d generated by K has similar dif...

Differentiability of Functions of MatricesMatrix Functions DIFFERENTIABILITY OF FUNCTIONS OF MATRICES

A. Let f be a function on the set of diagonal n × n matrices, and let˜f be the unique extension of f to the set of symmetric n × n matrices invariant with respect to conjugation by orthogonal matrices. We show that˜f has the same regularity properties as f.

Differentiability in Local Fields

Differentiability of Functions of Contractions

In this paper we study differentiability properties of the map T 7→ φ(T ), where φ is a given function in the disk-algebra and T ranges over the set of contractions on Hilbert space. We obtain sharp conditions (in terms of Besov spaces) for differentiability and existence of higher derivatives. We also find explicit formulae for directional derivatives (and higher derivatives) in terms of doubl...