Multiple Criteria Problems over Minkowski Balls

TL;DR

This paper investigates vector optimization problems involving Minkowski balls, focusing on balancing conflicting goals such as maximizing volume while minimizing width, within the space of symmetric convex compact sets.

Contribution

It introduces a framework for solving multi-criteria optimization problems over Minkowski balls, addressing conflicting objectives in convex geometry.

Findings

Developed methods for optimizing multiple criteria simultaneously

Provided solutions for balancing volume and width in convex bodies

Extended convex geometric optimization techniques

Abstract

Under study are some vector optimization problems over the space of Minkowski balls, i.e., symmetric convex compact subsets in Euclidean space. A typical problem requires to achieve the best result in the presence of conflicting goals; e.g., given the surface area of a symmetric convex body $\mathfrak x$, we try to maximize the volume of $\mathfrak x$ and minimize the width of $\mathfrak x$ simultaneously.