Logarithmically concave functions and sections of convex sets in $R^{n}$

Convex and concave relaxations of implicit functions

Convex Sets and Convex Functions

In this section, we introduce one of the most important ideas in economic modelling, in the theory of optimization and, indeed in much of modern analysis and computatyional mathematics: that of a convex set. Almost every situation we will meet will depend on this geometric idea. As an independent idea, the notion of convexity appeared at the end of the 19 century, particularly in the works of M...

Short Proofs of the Separation Theorems for L-convex/concave and M-convex/concave Functions

Recently K. Murota has introduced concepts of L-convex function and Mconvex function as generalizations of those of submodular function and base polyhedron, respectively, and has shown separation theorems for L-convex/concave functions and for M-convex/concave functions. The present note gives short alternative proofs of the separation theorems by relating them to the ordinary separation theore...

On convex-concave perception based functions

Perception based function (PBF) is given by the set of rules Ri:“If X is Ti then Y is Si”, where Ti is a linguistic term describing some fuzzy intervals Ai on the domain of real values of X and Si is a linguistic description of the shape of the function Y(X) on this interval. The methods of explicitation of such type of rules when Si are given by words QUICKLY INCREASING AND SLIGHTLY CONVEX, SL...

Multiple Positive Solutions for Semilinear Elliptic Equations in RN Involving Concave-Convex Nonlinearities and Sign-Changing Weight Functions

and Applied Analysis 3 Theorem 1.1. Assume that (A1) and (B1) hold. If λ ∈ 0,Λ0 , then Eλa,b admits at least one positive solution inH1 R . Associated with Eλa,b , we consider the energy functional Jλa,b inH1 R : Jλa,b u 1 2 ‖u‖H1 − λ q ∫ RN a x |u|dx − 1 p ∫