Abstract
We introduce the notion of strongly \(t\)-convex set-valued maps and present some properties of it. In particular, a Bernstein–Doetsch and Sierpinski-type theorems for strongly midconvex set-valued maps, as well as a Kuhn-type result are obtained. A representation of strongly \(t\)-convex set-valued maps in inner product spaces and a characterization of inner product spaces involving this representation is given. Finally, a connection between strongly convex set-valued maps and strongly convex sets is presented.
Highlights
Let (X, · ) be a normed space, D be a convex subset of X and let c > 0
We say that a set-valued map F : D → n(Y ) is strongly t-convex with modulus c if t F(x1) + (1 − t)F(x2) + ct (1 − t) x1 − x2 2 B ⊂ F(t x1 + (1 − t)x2), (2)
We prove that a set-valued map F is strongly convex with modulus c if and only if G = F +c · 2 B is convex. Using this condition, we obtain a new characterization of inner product spaces among normed spaces
Summary
Let (X, · ) be a normed space, D be a convex subset of X and let c > 0. Huang [11,12] extended the definition (1) of strongly convex functions to set-valued maps. He used such maps to investigate error bounds for some inclusion problems with set constraints. In this note we introduce the notion of strongly t-convex set-valued maps (strongly midconvex, in particular) and present some properties of it. Our definition is weaker than that given by Huang, but is easier to verify than that one since we assume that relation (2) below holds for only one fixed t ∈ (0, 1), not for all t ∈ (0, 1) It seems to be interesting and important that under weak regularity assumptions the class of strongly t-convex set-valued maps coincides with the class of strongly convex set-valued maps.
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