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Paraconvex, but not strongly, Takagi functions

Tabor J., Tabor J.

Control and Cybernetics

2012

Vol. 41, no. 3

545-559

There is an important open problem in the theory of approximate convexity whether every paraconvex function on a bounded interval is strongly paraconvex. Our aim is to show that this is not the case. To do this we need the following generalization of Takagi function. For a sequence a = (ai)i∈N ⊂ R+ we consider Takagi-like function of the form T(a)(x) := ∑ ∞ i=1 aidist(x, 12i-1Z) for x ∈ R. We give convenient conditions for verification whether T(a) is paraconvex or strongly paraconvex. This enables us to construct a class of paraconvex functions which are not strongly paraconvex.

Generalized approximate midconvexity

Tabor J., Tabor J.

Control and Cybernetics

2009

Vol. 38, no 3

655-669

Let X be a normed space and V ⊂ X a convex set with nonempty interior. Let α : [0, ∞) → [0, ∞) be a given nondecreasing function. A function ƒ : V → R is α(⋅)-midconvex if ƒ [wzór]. In this paper we study α(⋅)-midconvex functions. Using a version of Bernstein-Doetsch theorem we prove that if ƒ is α(⋅)-midconvex and locally bounded from above at every point then ƒ(rx + (1 - r)y) ≤ rƒ(x) + (1 - r)ƒ(y) + Pα(r, ¦¦x - y¦¦) for x,y ∈ V and r ∈ [0,1], where Pα : [0,1] x [0,∞) → [0,∞) is a specific function dependent on α. We obtain three different estimations of Pα. This enables us to generalize some results concerning paraconvex and semiconcave functions.