Convexity is one of the most important concepts in a study of analysis. Especially, it has been applied around the optimization problem widely. Our purpose is to define the concept of convexity of a set on Mizar, and to develop the generalities of convex analysis. The construction of this article is as follows: Convexity of the set is defined in the section 1. The section 2 gives the definition...

A convex non-convex variational model is proposed for multiphase image segmen-tation. We consider a specially designed non-convex regularization term which adapts spatially tothe image structures for better controlling of the segmentation and easy handling of the intensityinhomogeneities. The nonlinear optimization problem is efficiently solved by an Alternating Direc-tions Meth...

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...