Uniform convexity and the splitting problem for selections

Abstract

We continue to investigate cases when the Repov\v{s}-Semenov splitting problem for selections has an affirmative solution for continuous set-valued mappings. We consider the situation in infinite-dimensional uniformly convex Banach spaces. We use the notion of Polyak of uniform convexity and modulus of uniform convexity for arbitrary convex sets (not necessary balls). We study general geometric properties of uniformly convex sets. We also obtain an affirmative solution of the splitting problem for selections of certain set-valued mappings with uniformly convex images

Local characterization of strongly convex sets

“…Consider the family of short arcs A (p,q) of radius r joining points p, q ∈ A such that p − q = ε j and A (p,q) A. Hence, we see that 1). The family of short arcs considered previously also shows that p t and q t are interior points of Ω ∩ conv A relative to the subspace topology on conv A.…”

“…We note that once the existence of the limit has been established, its finiteness is also implied by the following result from [1], which holds in the more general setting of a uniformly convex Banach space X: If Ω ⊆ X is closed and uniformly convex, and Ω = X, then δ Ω (ε) ≤ C · ε 2 for some C ∈ R and all sufficiently small ε > 0.…”

“…The following related result has recently been established by Balashov and Repovš [2]: A closed, convex and bounded subset Ω of the Hilbert space X is r-convex, r > 0, if and only if there exists some ε > 0 such that condition (1) holds for all points x, y ∈ Ω whose distance does not exceed ε. Moreover, the property of strong convexity of a set Ω has also been characterized in terms of its modulus of convexity δ Ω [12],…”

On the Polyak convexity principle and its application to variational analysis

“…(iv) For every uniformly convex set S, a constant β > 0 can be proved to exist such that δ S (ǫ) ≤ βǫ 2 , ∀ǫ ∈ (0, diam S) (see [2]). Thus, a modulus of convexity of the power 2 is a maximal one.…”

Polyhedral approximations of strictly convex compacta

“…We proved in [3] that every uniformly convex set is bounded and if the Banach space E contains a nonsingleton uniformly convex set then it admits a uniformly convex equivalent norm. We also proved that the function ε → δ A (ε)/ε is increasing (see also [6, Lemma 1.e.8]), and for any uniformly convex set A there exists a constant C > 0 such that…”