On Approximately Midconvex Functions

Abstract

A real-valued function f defined on an open, convex set D of a real normed space is called (ε, δ)-midconvex if it satisfies fx + y 2The main result of the paper states that if f is locally bounded from above at a point of D and is (ε, δ)-midconvex, then it satisfies the convexity-type inequalityThe particular case ε = 0 of this result is due to Ng and Nikodem (Proc. Amer. Math. Soc. 118 (1993) 103-108), while the specialization ε = δ = 0 yields the theorem of Bernstein and Doetsch (Math. Ann. 76 (1915) 514-526).

On approximate t-convexity

“…A real valued function f defined on an open convex set D is called (ε, δ, p, t) -convex if it satisfiesThe main result of the paper states that if f is locally bounded from above at a point of D and (ε, δ, p, t) -convex (where 0 p < 1 and t 1/2 ) then it satisfies the convexity-type inequalitywhere ϕ :In the case p = 1, t = 1/2 analogous results were obtained in [2]. …”

On the stability of t-convex functions

“…Then the operators T 1,1/2 and S 1,1/2 are identical and their fixed point can be explicitly computed in the terms of the so-called Takagi function. Thus the following result can be obtained ( [3]): Corollary 1. Let ε 0 , ε 1 be nonnegative constants.…”

Approximate convexity and an edge-isoperimetric estimate

“…, x m ∈ [0, 1] with min i x i = x 1 and max i x i = x m . These classes were introduced in [L], except that the functions from the class F 2 (up to the normalization (2)) are known as (1, 1)-midconvex and under this name have been studied in a number of papers; see, for instance, [HP04,TT09b]. As shown in [L,Lemma 3], every concave function from the class F 0 belongs to all classes F m .…”