Multiobjective problems of convex geometry

Abstract

Under study is the new class of geometrical extremal problems in which it is required to achieve the best result in the presence of conflicting goals; e.g., given the surface area of a convex body x, we try to maximize the volume of x and minimize the width of x simultaneously. These problems are addressed along the lines of multiple criteria decision making. We describe the Pareto-optimal solutions of isoperimetric-type vector optimization problems on using the techniques of the space of convex sets, linear majorization, and mixed volumes.

Math-Selfie

“…D. Scott foresaw the role of Boolean valued models in mathematics and wrote as far back as in 1969: 17 We must ask whether there is any interest in these nonstandard models aside from the independence proof; that is, do they have any mathematical interest? The answer must be yes, but we cannot yet give a really good argument.…”

Multiple criteria problems over Minkowski balls

“…By way of example, consider a few multiple criteria problems of isoperimetric type. For more detail, see [4].…”

“…So it seems reasonable to suggest attractive theoretical problems that involve many criteria. Some geometrical problems of the sort were considered in [4]. In this article we address similar problems over symmetric convex bodies, using the the same technique that stems from the classical Alexandrov's approach to extremal problems of convex geometry [5].…”

Alexandrov's Approach to the Minkowski Problem

“…By way of example, consider a few multiple criteria problems of isoperimetric type. For more detail, see [14].…”