Takagi functions and approximate midconvexity

Tabor, Jacek

doi:10.1016/j.jmaa.2009.03.053

Cited by 26 publications

(48 citation statements)

References 12 publications

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“…, 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 .…”

Section: Summary Of Resultsmentioning

confidence: 99%

Approximate convexity and an edge-isoperimetric estimate

Lev

2014

Journal of Mathematical Analysis and Applications

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Abstract. We study extremal properties of the functionwhere x = min{x, 1 − x}. In particular, we show that F is the pointwise largest function of the class of all real-valued functions f defined on the interval [0, 1], and satisfying the relaxed convexity conditionand the boundary condition max{f (0), f (1)} ≤ 0.As an application, we prove that if A and S are subsets of a finite abelian group G, such that S is generating and all of its elements have order at most m, then the number of edges from A to its complement G \ A in the directed Cayley graph induced by S on G is

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Section: Summary Of Resultsmentioning

confidence: 99%

Approximate convexity and an edge-isoperimetric estimate

Lev

2014

Journal of Mathematical Analysis and Applications

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“…Note that for p = 0 (1.1) reduces to the estimation obtained by C. T. Ng and K. Nikodem. The case p ∈ [1,2] was completely solved in [12], where the authors showed that the optimal estimation for the convexity differences of (ε, p)-midconvex functions is given by…”

Section: Introductionmentioning

confidence: 99%

Numerical verification of condition for approximately midconvex functions

Spurek

Tabor

2012

Aequat. Math.

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Let X be a normed space and V be a convex subset of X. Let α :It can be shown that every continuous α-midconvex function satisfies the following estimation:It is an important problem to verify for which functions α the above estimation is optimal. The conjecture of Páles that this is the case for functions of type α(r) = r p for p ∈ (0, 1), was proved by Makó and Páles (J Math Anal Appl 369:545-554, 2010). In this paper we present a computer assisted method to verify the optimality of this estimation in the class of piecewise linear functions α.Mathematics Subject Classification (2010). Primary 26A51; Secondary 39B62, 65G40.

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“…An important issue in these papers is to obtain estimations of the bounds which appear for approximately convex functions, see, for example, Cannarsa and Sinestrari [1], Ng and Nikodem [15], Rolewicz [17][18][19]. It also turned out that Takagi-like functions appear naturally in the investigation of approximate convexity, see, for example, Házy [5], [6], Házy and Páles [7][8][9], Makó and Páles [11][12][13], Mureńko, Tabor and Tabor [14], Tabor and Tabor [20,21], Tabor, Tabor andŻo ldak [22].…”

Section: Introductionmentioning

confidence: 99%

“…Makó and Páles [12,13], Tabor and Tabor [20,21]). If (3) holds for all x, y ∈ D and for all t ∈ [0, 1] then we say that f is ϕ-convex.…”

Section: Introductionmentioning

confidence: 99%