INFORMATION TO USERS This manuscript has been reproduced from the microfilm master. UMI films the text directly from the original or copy submitted. Thus, some thesis and dissertation copies are in typewriter face, while others may be from any type of computer printer. The quality of this reproduction is dependent upon the quality of the copy submitted. Broken or indistinct print, colored or poor quality illustrations and photographs, print bleedthrough, substandard margins, and improper alignment can adversely affect reproduction. In the unlikely event that the author did not send UMI a complete manuscript and there are missing pages, these will be noted. Also, if unauthorized copyright material had to be removed, a note will indicate the deletion. Oversize materials (e.g., maps, drawings, charts) are reproduced by sectioning the original, beginning at the upper left-hand comer and continuing from left to right in equal sections with small overlaps. Each original is also photographed in one exposure and is included in reduced form at the back of the book. Photographs included in the original manuscript have been reproduced xerographically in this copy. Higher quality 6” x 9” black and white photographic prints are available for any photographs or illustrations appearing in this copy for an additional charge. Contact UMI directly to order. UMI A Bell & Howell Information Company 300 North Zeeb Road, Ann Aibor MI 48106-1346 USA 313/761-4700 800/521-0600 SYMMETRIZATIONS AND CONVOLUTIONS OF CONVEX BODIES DISSERTATION Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of The Ohio State University By Antonis Tsolomitis, B.Sc. **** The Ohio State University 1996 Dissertation Committee: Approved by W. Davis, Advisor G. Edgar / / Advisor V. Milman Department of Mathematics M. Talagrand UMI Number: 9630994 UMI Microform 9630994 Copyright 1996, by UMI Company. All rights reserved. This microform edition is protected against unauthorized copying under Title 17, United States Code. UMI 300 North Zeeb Road Ann Arbor, MI 48103 to my parents who gave me life and to my teachers who taught me how to live it ii AKN OWLEDGEMENTS I express deep appreciation to Professor V. Milman not only for his guidance and insight throughout the research but also for showing me the philosophical aspects of Mathematics through Wittgenstein’s assertion: Mathematics is a lan­ guage ([27]). Gratitude is expressed to Professor W. Davis for without him this dissertation would not have been possible. Thanks go to the other members of my advisory committee, Professors G. Edgar and M. Talagrand, for their suggestions and comments. I offer sincere thanks to Professor W. Johnson for making my visit to Texas A&M University possible. The technical assistance of Dr. G. Bhatnagar, V. Gougoulides and P. Zafiropoulos is gratefully acknowledged. I offer sincere thanks to the communities of the Hellenic Students at the Ohio State University and Texas A&M University for their support and especially to E. Mavromatidou, Dr. C. Rouvas and Z. Zeniou. Special thanks go to A. Apostolopoulos and E. Mavromatidou for the philosophical discussions we frequently had. To my family in Greece, I thank you for your transatlantic support and humor. VITA June 25, 1967 ........................................................ Born, Piraeus, Greece 1989 .........................................................................B.Sc., University of Athens, Athens, Greece 1989-1995 .................................................................Graduate Teaching Associate, Department of Mathematics, The Ohio State University, Columbus, Ohio 1995-1996 ............................................................... Presidential Fellow, The Ohio State University, Columbus, Ohio FIELDS OF STUDY Major Field : Mathematics iv TABLE OF CONTENTS DEDICATION ..................................................................................................................ii ACKNOWLEDGMENTS ..............................................................................................iii VITA ..................................................................................................................................iv TABLE OF CONTENTS ............................................................................................... v LIST OF FIGURES........................................................................................................vi CHAPTER I. Introduction .............................................................................................1 II. B ackground ............................................................................................ 12 III. Symmetrizations O f C onvex B odies ....................................... 27 IV. C onvolution O f C onvex B odies ..................................................56 REFERENCES............................................................................................................... 92 LIST OF FIGURES FIGURES 1. Lower bound of the symmetrized body K ...................................................50 2. Lower bound of v.rad.(K OF) ...................................................................... 54 vi CHAPTER I INTRODUCTION 1 .Prolegomena Geometry is one of those elements of human civilization whose value can not be overestimated. Looking back in the history of science we find geometry to be its very foundation. Starting from the need to “measure the earth”-as its name reveals (geometry < 'yea + tierpu = earth + to measure)~it was the first discipline to become into a rigorous science. It was through Geometry and Philosophy that the purely abstract thinking was developed and the ideas of theory and proof were born. Consequently geometry is in the basis of practically everything. From the construction of the musical scales by Pythagoras, to the way we understand our universe after its geometric description by Aristarchus. From something as theoretical as physics to something as applied as medicine. Thus it would not be an exaggeration to say that Geometry is one of the fundamental elements of the history of human civilization. 1 One of the problems that appeared in early geometry was what is today called “the isoperimetric problem”. The problem asks to find that closed plane curve of given length, that encloses the maximum possible volume and the circle is the solution. The problem made its appearance in antiquity and information about arguments to its favor have reached us through the work of Zenodor (2nd century B.C.; see [5]). The first rigorous proof for the circle being the solution of this problem, was given in the year 1882 by Edler. It is not difficult to see that if the maximum of enclosed area is achieved by some curve, then the enclosed region must be a convex set, i.e. the intersection of this region with any line of the plane consists only of one line segment. One of the ways to verify the solution of the isoperimetric problem is to use the Steiner symmetrization. To symmetrize a segment of finite length on the plane with respect to a perpendicular to the segment line, means to move the segment in the direction vertical to the line until it is bisected by the line. Let K be a region in the plane enclosed by a curve so that it is convex. Let £ be a line in the plane. Symmetrizing with respect to I the intersections of K with all lines perpendicular to £, we arrive at a new body K , the Steiner symmetrization of K. This procedure preserves the volume but increases the length of the boundary of K. Moreover we can keep symmetrizing with respect to certain lines, so that the resulting bodies “converge” to a Euclidean ball. The sense of this convergence will be defined rigorously in the next chapter. Thus if we want to keep the length of the boundary fixed, the volume must be increased and since the symmetrizations lead to a Euclidean ball, that must be the solution to the isoperimetric problem. Passing now to higher dimensional spaces and understanding their geometry is something that science was and still is in need for. The need of many dimen­ sions does not stem from mathematical curiosity only (although this would be considered by many philosophers and mathematicians a complete justification of the need) but also from the needs of applied science and theoretical physics. For example the mathematical basis of quantum mechanics is the theory of linear op­ erators in a Hilbert space. The success of Hilbert spaces in such applications lies on the fact that their geometry is “essentially Euclidean” and hence we have a very good understanding of them. Since non-Hilbert spaces appear naturally in applications it seems necessary to study the geometry of these spaces. The difficulties that appear in this study come from the existence of many counter-intuitive phenomena that take place in “high” dimensional spaces. One of these, in the language developed by V.D.Milman, is the “concentration of measure phenomena”: let e > be given. Cut out from the n-dimensional Euclidean sphere in Rn two cups around the north and south poles of the sphere using two parallel hyperplanes being at distance e. What remains is a “band” around the equator of height e. The surface area of this band converges exponentially fast to the surface area of the entire sphere as the dimension n increases to infinity (one should normalize the measure on the sphere so that its surface area equals 1). This means that in high dimension the surface measure on a sphere concentrates strongly