Minisum Hyperspheres

ISBN-13: 9781441998064 / Angielski / Twarda / 2011 / 116 str.

Mark-Christoph Korner; Mark-Christoph K. Rner

Opis

This book presents a self-contained introduction to the theory of minisum hyperspheres. This specialized research area within the larger field of geometric optimization is full of interesting and open problems.This work provides an overview of the history of minisum hyperspheres as well as describes the best techniques for developing and solving minisum hypersphere problems. Various related areas of geometric and nonlinear optimization are also discussed.As the first publication devoted to this area of research, this work will be of great interest to graduate-level researchers studying minisum hypersphere problems as well as mathematicians interested geometric optimization.

Recenzje

From the reviews:

"This little book is fully devoted to what is known about the following median hypersphere problem and several of its variants ... . This book will be of interest to any researcher interested in continuous location theory and/or distance geometry." (Frank Plastria, Mathematical Reviews, Issue 2012 f)

Spis treści

-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.

Blurb

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.