Topics in Hyperplane Arrangements, Polytopes and Box-Splines
Several mathematical areas that have been developed independently over the last 30 years are brought together revolving around the computation of the number of integral points in suitable families of polytopes. The problem is formulated here in terms of p
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Corrado De Concini • Claudio Procesi
Topics in Hyperplane Arrangements, Polytopes and Box-Splines
Corrado De Concini Università di Roma “La Sapienza” Dipartimento di Matematica Piazzale Aldo Moro 5 00185 Roma Italy [email protected]
Claudio Procesi Università di Roma “La Sapienza” Dipartimento di Matematica Piazzale Aldo Moro 5 00185 Roma Italy [email protected]
Editorial Board: Sheldon Axler, San Francisco State University Vincenzo Capasso, Università degli Studi di Milano Carles Casacuberta, Universitat de Barcelona Angus MacIntyre, Queen Mary, University of London Kenneth Ribet, University of California, Berkeley Claude Sabbah, CNRS, École Polytechnique Endre Süli, University of Oxford ´ Wojbor Woyczynski, Case Western Reserve University
ISBN 978-0-387-78962-0 e-ISBN 978-0-387-78963-7 DOI 10.1007/978-0-387-78963-7 Springer New York Dordrecht Heidelberg London Library of Congress Control Number: 2010934230 © Springer Science+Business Media, LLC 2010 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)
Contents
Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xix Part I Preliminaries 1
Polytopes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Convex Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.1 Convex Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.2 Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.3 Lines in Convex Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.4 Faces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Polyhedra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.1 Convex Polyhedra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.2 Simplicial Complexes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.3 Polyhedral Cones . . . .
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