Minisum Hyperspheres
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 distan
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Aims and Scope Optimization has been expanding in all directions at an astonishing rate during the last few decades. New algorithmic and theoretical techniques have been developed, the diffusion into other disciplines has proceeded at a rapid pace, and our knowledge of all aspects of the field has grown even more profound. At the same time, one of the most striking trends in optimization is the constantly increasing emphasis on the interdisciplinary nature of the field. Optimization has been a basic tool in all areas of applied mathematics, engineering, medicine, economics, and other sciences. The series Springer Optimization and Its Applications publishes undergraduate and graduate textbooks, monographs, and state-of-the-art expository work that focus on algorithms for solving optimization problems and also study applications involving such problems. Some of the topics covered include nonlinear optimization (convex and nonconvex), network flow problems, stochastic optimization, optimal control, discrete optimization, multiobjective programming, description of software packages, approximation techniques and heuristic approaches.
For further volumes: http://www.springer.com/series/7393
Mark-Christoph K¨orner
Minisum Hyperspheres
ABC
Mark-Christoph K¨orner Institut f¨ur Numerische und Angewandte Mathematik Georg-August-Universit¨at G¨ottingen G¨ottingen 37083 Germany [email protected]
ISSN 1931-6828 ISBN 978-1-4419-9806-4 e-ISBN 978-1-4419-9807-1 DOI 10.1007/978-1-4419-9807-1 Springer New York Dordrecht Heidelberg London Library of Congress Control Number: 2011929987 Mathematics Subject Classification (2010): 46B20, 52A21, 90B85 c Springer Science+Business Media, LLC 2011 All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)
Preface
A mathematical problem should be difficult in order to entice us, yet not completely inaccessible, lest it mock at our efforts. (David Hilbert)
The most fascinating mathematical problems share two properties. On the one hand, they are easy to understand and can be expressed in simple terms. On the other hand, they are difficult to solve. Probably the best-known example for such a problem is Fermat’s Last Theorem: No three positive integers a, b, and c can satisfy the equation an + bn = cn for any integer v
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