Gabriela Cristescu

Non-Connected Convexities and Applications

Table of contents. Preface. Acknowledgements. Main notations. Part 1: Non-connected convexity properties. 1. The fields of non-connected convexity properties. 2. Convexity with respect to a set. 3. Behaviours. Convexity with respect to a behaviour. 4. Convexity with respect to a set and two behaviours. 5. Convexities defined by means of distance functions. 6. Induced convexity. 7. Convexity defined by means of given functions. 8. Classification of the convexity properties. Part 2: Applications. 9. Applications in pattern recognition. 10. Alternative theorems and integer convex sets. 11. Various types of generalised convex functions and main properties. 12. Applications in optimisation. 13. Applications in pharmaco-economics. References. Authors index. Subject index. Figures index. Tables index.

Classes of Discrete Convexity Properties

Abstract In the general first-level classification of the convexity properties for sets, discrete convexities appear in more classes. A second-level classification identifies more subclasses containing discrete convexity properties, which appear as approximations either of classical convexity or of fuzzy convexity. First, we prove that all these convexity concepts are defined by segmental methods. The type of segmental method involved in the construction of discrete convexity determines the subclass to which it belongs. The subclasses containing the convexity properties that have discrete particular cases are also presented.

Bounds having Riemann type quantum integrals via strongly convex functions

The aim of this paper is to obtain some new bounds having Riemann type quantum integrals within the class of strongly convex functions. The results obtained are sharp on limit q → 1. These new results reduce to Tariboon-Ntouyas, Merentes-Nikodem and other previously known results when q → 1, where 0 < q < 1. The sharpness of the results of Tariboon-Ntouyas and Merentes-Nikodem is proved as a consequence.

Applications in pattern recognition

Many approaches of the convexity property for digital sets (sets of pixels) are known, having applications in various algorithms for the detection of the convexity of a configuration appearing in an image. All of them describe convexity for a non-connected set (set of pixels), which is finite and is embedded into R 2 or R 3 . Another purpose of the study of the discrete description of the convexity is to find a possibility of measuring the “degree of convexity” of a set in order to obtain accuracy in digital imagery. This is very important because these images might represent parts of the inside of the human body that are often investigated by physicians for diagnosis. In this chapter we will deal with both aspects. We shall discuss the connection between the convexity properties presented in chapter 5 of this book and the possibility of measuring the sizes of concavities of a set and of detecting them. Also, we shall present the way in which the detection of the convexity property appears in cytology.

Some inequalities for functions having an s-convex derivative of superior order

Some Hermite-Hadamard type inequalities via fractional integration are derived for superior order differentiable functions having one derivative with s -convexity of either first kind or second kind. The n -th order cumulative behavior of the function in the neighborhood of the frontier of the definition interval is studied in case of the s -convexity of second kind, by means of fractional integration. The inequalities are as best as possible from the sharpness point of view, meaning that a sharpness class of functions is identified, for each inequality, within the functions that have one derivative that is s -affine either of first kind or of second kind. Mathematics subject classification (2010): 26A33, 26D15, 26A51.

Some inequalities for functions having Orlicz-convexity

Some Hermite-Hadamard type inequalities are derived for products of functions having Orlicz-convexity properties. We also obtain these inequalities via Riemann-Liouville fractional integrals for Orlicz-convex functions. These inequalities are as best as possible from the sharpness point of view, meaning that a sharpness class of functions is identified, for each inequality, within the functions that are s-affine of first kind. Some special cases are discussed.

Integral inequalities for (Orlicz, Breckner) co-ordinated convex functions on rectangles

Abstract The concept of (Orlicz, Breckner) co-ordinated (s, σ)-convexity is introduced for two variables real functions. Upper and lower sharp integral boundary properties of Hermite-Hadamard type are proved on rectangles.

Regularity properties and integral inequalities related to (k; h1; h2)-convexity of functions

Abstract The class of (k; h1; h2)-convex functions is introduced, together with some particular classes of corresponding generalized convex dominated functions. Few regularity properties of (k; h1; h2)-convex functions are proved by means of Bernstein-Doetsch type results. Also we find conditions in which every local minimizer of a (k; h1; h2)-convex function is global. Classes of (k; h1; h2)-convex functions, which allow integral upper bounds of Hermite-Hadamard type, are identified. Hermite-Hadamard type inequalities are also obtained in a particular class of the (k; h1; h2)- convex dominated functions.

Restricted Oriented Slack Convexity and Extremal Elements with Respect to a Set

Abstract. In a real linear space V , a set of directions O ⊂ V and a set of representative vectors M ⊂ V generate a double re- stricted concept of convexity: with directional and with particular punctual restrictions. A Krein-Milman type property for slack O- directionally convex sets with respect to M is derived in terms of double restricted extremal elements of a set.

Bi‐Level Generalized Quasi‐Convex Optimization in Ecologic‐Economic Efficiency

The concepts of g‐transformed convex set and the related g‐transformed quasi‐convex function are defined and studied in connection with the problem of assessing the ecologic‐economic efficiency of an investment or development policy. A bi‐level g‐transformed quasiconvex problem is proposed to approach the efficiency in terms of general welfare. The existence of the solution is studied, together with its geometric position in the space of admissible solutions.