Integral Geometry And Convexity: Proceedings of the International Conference, Wuhan, China, 18 - 23 October 2004 PDF
Preview Integral Geometry And Convexity: Proceedings of the International Conference, Wuhan, China, 18 - 23 October 2004
Proceedings of the International Conference Editors World Scientific INTEGRAL GEOMETRY AND CONVEXITY This page is intentionally left blank Proceedings of the International Conference INTEGRAL GEOMETRY AND CONVEXITY Wuhan, China 18 - 23 October 2004 Editors Eric L Grinberg University of New Hampshire, USA Shougui Li Wuhan University of Sciences and Technology, China Gaoyong Zhang Polytechnic University, USA Jiazu Zhou Guizhou Normal University, China \jjp World Scientific NEWJERSEY • LONDON • SINGAPORE • BEIJING • SHANGHAI • HONGKONG • TAIPEI • CHENNAI Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library. INTEGRAL GEOMETRY AND CONVEXITY Proceedings of the International Conference Copyright © 2006 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher. For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher. ISBN 981-256-513-2 Printed in Singapore by World Scientific Printers (S) Pte Ltd PREFACE Integral geometry, known as geometric probability in the past, origi nated from BufFon's needle experiment. Crofton, Poincare, Blashke, Chern, Santalo and others made significant contributions in the past centuries. The applications of this field vary from the medical sciences to other mathematic branches including algebra, geometric inequalities, differential equations, topology, and geometric convexity. Remarkable advances have also been made in several areas that involve the theory of convex bodies. In 2002, a select group of mathematicians believed it was the time to convene the first conference on these important fields in China. The result was the First International Conference on Integral Geometry and Convexity Related Topics in China, which was held at Wuhan Univer sity of Science and Technology from October 18 to 23, 2004 and sponsored by the Mathematical Associations of Hubei Province and Wuhan City. The organizers carefully selected international researchers who are leaders in their specialties to discuss their recent results and their ideas on the trends of future research. The only regret is that some known mathematicians could not present due to their own schedule. The program consisted of talks on integral geometry, convex geome try, complex geometry, probability, statistics and other convexity-related branches. The principal speakers were Rolf Schneider, Eric Grinberg, and Ralph Howard; they were joined by ten other leading international mathematicians and eight domestic Chinese researchers. The conference was purposely designed to facilitate the discussion and exchange of ideas among researchers with various specialties and we believe that objective was achieved. It is especially gratifying to have the talks, which were presented at the Conference collected together in these Proceedings (World Scientific Publishers, Singapore). The major themes include probabilistic and ana lytic methods in the study of convex bodies, especially in high dimensions, applications of integral geometry and convex-geometric methods to other branches of mathematics, isoperimetric-type inequalities, radon transforms, and applications to medical, economic and information sciences. The ed- V vi itors of the Proceedings are Jiazu Zhou, Gaoyong Zhang, Eric Grinberg and Shougui Li, all of whom should also be recognized for their hard work and diligence in the planning and execution of all aspects of this inter national conference. Since I was unable to attend the conference due to physical mobility difficulties, I am looking forward to the publication of the Proceedings. On behalf of the organizing committee, we would also like to express our sincere gratitude to the invited speakers and audience of over 50 research mathematicians whose participation made this conference a great success. We hope that this conference is the start of many more such gatherings. Chuan Chih Hsiung Lehigh University June, 2005 FORWARD For several decades students in Wuhan were introduced to and mentored in the subjects of integral geometry, convexity, geometric inequalities, ge ometric probability and allied fields under the direction of Ren Delin and with the support of S.S. Chern and C.C. Hsiung. Indeed, this mathemati cal locale provided unique opportunities for students to learn about these beautiful and important subjects, which might otherwise have been quite inaccessible to these future mathematicians. Wuhan graduates went on to make broad and substantial research contributions. Thus it seemed par ticularly appropriate to hold a conference on these topics in Wuhan City and Hubei Province, a special place for integral and convex geometry, and in a special year, when Professor Ren Delin was about to retire at age 70. Some participants, myself included, were fortunate to have visited Wuhan in earlier decades and hence to witness the vast and rapid development of academic facilities and growth of support of the mathematical sciences. The conference infrastructure was superb. The local organizers are sin cerely thanked for the outstanding environment which they crafted for the event and for the great care they took each day to insure that all had what was needed. We also thank the National Science Foundation which supported some of the organizers through USA-China Cooperative grant DMS-9906856. The community of participants was lively, with broad representation which made a stimulating atmosphere for the exchange of ideas, and which is reflected in the proceedings. All regretted that medical travel restrictions prevented S.S. Chern from attending the event. However, Professor Chern sent a letter of support to the conferees which was read during the opening ceremonies, and his support was felt throughout. It was most saddening to learn of his untimely death in December, 2004. At the same time, it is fortunate that the conference was held during Professor Chern's lifetime. For many conferees it provided the last link to a great mathematician and a great man. Eric L. Grinberg University of New Hampshire vii This page is intentionally left blank CONTENTS Preface v Foreword vii Volume inequalities for sets associated with convex bodies 1 Stefano Campi and Paolo Gronchi Integral geometry and Alesker's theory of valuations 17 Joseph H.G. Fu Area and perimeter bisectors of planar convex sets 29 Paul Goodey Radon Inversion: from lines to Grassmannians 37 Eric Grinberg Valuations in the affine geometry of convex bodies 49 Monika Ludwig Crofton measures in projective Finsler spaces 67 Rolf Schneider Random methods in approximation of convex bodies 99 Carsten Schiitt Some generalized maximum principles and their applications to Chern type problems 107 Young Jin Suh Floating bodies and illumination bodies 129 Elisabeth Werner Applications of information theory to convex geometry 141 Deane Yang Containment measures in integral geometry 153 Gaoyong Zhang and Jiazu Zhou
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