This volume presents a self-contained introduction to the theory of minisum hyperspheres. The minisum hypersphere problem is a generalization of the famous Fermat-Torricelli problem. The problem asks for a hypersphere minimizing the weighted sum of distances to a given point set. In the general framework of finite dimensional real Banach spaces, the minisum hypersphere problem involves defining a hypersphere and calculating the distance between points and hyperspheres. The theory of minisum hyperspheres is full of interesting open problems which impinge upon the larger field of geometric optimization.

This work provides an overview of the history of minisum hyperspheres as well as describes the best techniques for analyzing and solving minisum hypersphere problems. Related areas of geometric and nonlinear optimization are also discussed.

Key features of Minisum Hyperspheres include:

-assorted applications of the minisum hypersphere problem

• a discussion on the existence of a solution to the problem with respect to Euclidean and other norms

• several proposed extensions to the problem, including a highlight of positive and negative weights and extensive facilities extensions

This work is the first book devoted to this area of research and will be of great interest to graduate students and researchers studying the minisum hypersphere problems as well as mathematicians interested in geometric optimization.

Inhalt

-Preface.- 1. Basic Concepts (Circles and Hyperspheres, Minisum Hyperspheres, Mathematical Preliminaries, Finite Dominating Sets).- 2. Euclidean Minisum Hyperspheres (Basic Assumptions, Distance, Degenerated Solutions, Existence of Optimal Solutions, Incidence Properties, Solution Approaches for the Planar Case, Concluding Remarks).- 3. Minisum Hyperspheres in Normed Spaces (Basic Assumptions, Distance, Degenerated Solutions, Existence of Minisum Hyperspheres, Incidence Properties, Polyhedral Norms in the Plane, Concluding Remarks).- 4. Minisum Circle Problem with Unequal Norms (Basic Assumptions, Distance, Properties of Minisum Circles, Polyhedral Norms, Concluding Remarks).- 5. Minisum Rectangles in a Manhattan Plane (Basic Assumptions, Notations, Point-Rectangle Distance, Restricted Problems, Unrestricted Problem, Concluding Remarks).- 6. Extensions. Bibliiography. Index.