A distance function property implying differentiability

Abstract

In a real normed linear space X, properties of a non-empty closed set K are closely related to those of the distance function d which it generates. If X has a uniformly Gateaux (uniformly Frechet) differentiable norm, then d is Gateaux (Frechet) differentiable at x e X \ K if there exists an t f l , || "i* || = 1 such thatand is Gateaux (Frechet) differentiable on X \ K if there exists a set P+(K) dense in X\K where such a limit is approached uniformly for all x £ P^.(K). When X is complete this last property implies that K is convex.Given a real normed linear space X a non-empty closed subset K generates a distance function d on X where d{x) = inf{||x -T/ II : y 6 K}. A distance function d always satisfies the Lipschitz condition \d{x) -d{y)\ ^ \\x -y\\f o r a l l x,y € X and is a convex function if and only if if is a convex set.There is a natural duality between the properties of the set K and its distance function d. We see this duality exploited in abstract approximation theory and in techniques of non-smooth optimisation.A longstanding abstract approximation problem concerns Chebyshev sets. A subset K is said to be a Chebyshev set if for each x G X \ K there exists a unique p(x) € K such that <*(*) = H X -P O O H .The problem is to determine necessary and sufficient conditions for a Chebyshev set to be convex. The best result so far was given by Vlasov [10,11] who showed that in

On Closures of Preimages of Metric Projection Mappings in Hilbert Spaces

“…Hence by (19) we obtain s i ∈ S ∩B H [s, ] for i ∈ N large enough, which contradicts the choice of the sequence {s i } i∈N . Thus u ρ ∈ int D S (S) and consequently u ∈ cl int D S (S).…”

“…Another way to get the convexity of a Chebyshev set is to assume a differentiability of the distance function outside the set. Namely, if the distance function to a Chebyshev set is Fréchet differentiable at all points outside the set then the set is convex too, we refer to [13,14,18,19] for details. Due to L. P. Vlasov we know also that the continuity of the metric projection (which implies the convexity) can be obtained by checking if the Vlasov condition is satisfied, see (22) and [35, page 56], we refer also to [33,34].…”

Continuity characterisations of differentiability of locally Lipschitz functions

“…In [5,Corollary 2,p.64] it was shown that a similar result holds for a distance function d on a normed linear space X with uniformly Gateaux (uniformly Frechet) differentiable norm.…”

Differentiability of Distance Function and The Proximinal Condition implying Convexity

“…The following Proposition provides a necessary and sufficient condition for differentiability of the distance function if E ′ (K) is dense in X K. We do not assume the uniform differentiability conditions as in Giles [12,Proposition 2]. Though, in [13] the author has given a characterization for the smoothness (Fréchet smoothness) of distance function without assuming density of E(K) but under the uniform differentiability constraints. Hence, we hope, our result advances that of Giles [12,13].…”