On the boundedness and continuity of convex functions and additive functions

Aequationes mathematicae -     

On the boundedness, Christensen measurability and continuity of t-Wright convex functions

We discuss the connections between boundedness and continuity of t-Wright convex functions, moreover, we generalize some results of P. Fischer and Z. S?odkowski [4] concerning the Christensen measurability of Jensen convex functions to the case of t-Wright convex functions.     

On the continuity of biconjugate convex functions

We show that a Banach space is a Grothendieck space if and only if every continuous convex function on has a continuous biconjugate function on , thus also answering a question raised by S. Simons. Related characterizations and examples are given.

     

On the continuity of midpoint hull convex set-valued functions

Summary The concept of hull convexity (midpoint hull convexity) for set-valued functions in vector spaces is examined. This concept, introduced by A. V. Fiacco and J. Kyparisis (Journal of Optimization Theory and Applications,43 (1986), 95–126), is weaker than one of convexity (midpoint convexity).The main result is a sufficient condition for a midpoint hull convex set-valued function to be continuous. This theorem improves a result obtained by K. Nikodem (Bulletin of the Polish Academy of Sciences, Mathematics,34 (1986), 393–399).     

Boundedness and continuity of additive and convex functionals

Summary It is shown that for any real Baire topological vector spaceX the set classesA(X):={T : for any open and convex setD T, every Jensen-convex functional, defined onD and bounded from above onT, is continuous} andB(X):={T : every additive functional onX, bounded from above onT, is continuous} are equal. This generalizes a result of Marcin E. Kuczma (1970) who has shown the equalityA( ⁿ )=B( ⁿ ) However, the infinite dimensional case requires completely different methods; therefore, even in the caseX = ⁿ we obtain a new (and perhaps simpler) proof than that given by M. E. Kuczma.     

On rational approximation of convex functions with a given modulus of continuity

Доказывается, что для наименьших равномер ных рациональных уклоне нийR _n(f) выпуклой на [0,1] функции с модулем непрерывно сти, не превосходящемω(δ), сп раведлива оценка $$R_n (f) \leqq c\frac{{\ln ^2 n}}{{n^2 }}\mathop {\max }\limits_{e^{ - n} \leqq \theta< 1} \left\{ {\omega (\theta )\ln \frac{1}{\theta }} \right\},$$ гдес — абсолютная по стоянная.     

Differentiability of convex functions and the convex point of continuity property in Banach spaces

We show that ifX is a separable Banach space, then every continuous, convex, Gateaux differentiable function onX is Fréchet differentiable on a dense set if and only ifX* has theweak*-Convex Point of Continuity Property (C*PCP). Research completed while a visitor at the University of Alberta. Research supported in part by an H. R. MacMillan Fellowship from the University of British Columbia. Research partially supported by NSERC (Canada).     

Sum of graphs of continuous functions and boundedness of additive operators

Assume that and are uniformly continuous functions, where D₁,D₂⊂X are nonempty open and arc-connected subsets of a real normed space X. We prove that then either f and g are affine functions, that is f(x)=x^∗(x)+a and g(x)=x^∗(x)+b with some x^∗∈X^∗ and a,b∈R or the algebraic sum of graphs of functions f and g has a nonempty interior in a product space X×R treated as a normed space with a norm .     

On the continuity of functions

In this paper, results of a questionnaire about continuity of functions are analysed. The tested students attend Novi Sad grammar school. The aim of this test was to check the student's theoretical and visual knowledge of continuous functions at the end of their high school education. The questions were given with and without the graphs of functions. The main conclusion is that the graph of a function has a significant influence on the students opinion about continuity. Looking at the graphs of functions mostly led to correct answers, but sometimes it causes problems, if it has a break. The idea came from a 1981 paper by Tall and Vinner, where the “concept image” of a continuous functions was examined.     

On hyperbolically convex functions

Let be the unit disk of the complex plane. A conformai map of into itself is called hyperbolically convex if the non-Euclidean segment between any two points of also belongs to . In this paper we prove several inequalities that are analogous to inequalities about (Euclidean) convex univalent functions. We show that if ƒ (0) = 0, then Re zf′/f > 1/2. This inequality is the key for the results of this paper. In particular we deduce a three-variable inequality corresponding to that of Ruscheweyh and Sheil-Small. The sharp bound for the Schwarzian derivative remains open.     

On the singularities of convex functions

Given a semi-convex functionu: ω⊂R ⁿ→R and an integerk≡[0,₁,n], we show that the set ∑^k defined by

On uniformly convex functions

Non-convex functions that yet satisfy a condition of uniform convexity for non-close points can arise in discrete constructions. We prove that this sort of discrete uniform convexity is inherited by the convex envelope, which is the key to obtain other remarkable properties such as the coercivity. Our techniques allow to retrieve Enflo's uniformly convex renorming of super-reflexive Banach spaces as the regularization of a raw function built from trees. Among other applications, we provide a sharp estimation of the distance of a given function to the set of differences of Lipschitz convex functions. Finally, we prove the equivalence of several possible ways to quantify the super weakly noncompactness of a convex subset of a Banach space.     

On convex triangle functions

We prove that the strongest (largest convex) solution of the functional inequality $$\tau \left( {\frac{{F + G}}{2},\frac{{H + K}}{2}} \right) \le \frac{{\tau (F,H) + \tau (G,K)}}{2},$$ whereF, G, H andK are arbitrary distribution functions, is the triangle function τ(F, G)(x) = Max(F(x) +G(x) ? 1, 0).