Herbert Busemann. Selected Works. vol. II

Autor: Papadopoulos, Athanase
Přispěvatelé: Institut de Recherche Mathématique Avancée (IRMA), Université de Strasbourg (UNISTRA)-Centre National de la Recherche Scientifique (CNRS), Papadopoulos, Athanase
Jazyk: angličtina
Rok vydání: 2018
Předmět:
Zdroj: 2018, Herbert Busemann. Selected Works. vol. II, Springer Verlag, Cham, 2018, 860p., 978-3-319-65623-6
Popis: International audience; The present volume of Busemann’s Selected Works, the second of the twovolumes set, contains his papers on convexity and integral geometry, thoseon Hilbert’s Fourth Problem, and some other papers on various topics. Thepapers on convexity cover the period 1935–1969, whereas those on Hilbert’sProblem IV were published later, between 1966 and 1984.Busemann’s papers are preceded by introductory material including biographicalelements, correspondence, and five essays, by various authors,which may be considered as reading guides for his work.The correspondence is edited, presented and translated by Manfred Karbe.It is based on a collection of letters that are kept at the Elmer HolmesBobst Library of New York University, written by Herbert Busemann, AlfredBusemann, William Feller and Richard Courant. These letters give alively picture of the period 1933–1936 which Busemann spent in Denmark,including his doubts and hesitations concerning his project of emigratingto the US. The letters also provide interesting information on the scientists(mostly mathematicians, but also physicists) with whom Busemann interactedduring that period. Besides the letters, the present volume containsa biography of Herbert Busemann extracted from Anik´o Sz´abo’s Vertreibung,R¨uckkehr, Wiedergutmachung (Expulsion, return, reparation), WallsteinVerlag, G¨ottingen, 2000, related to his request to the German State ofa compensation for the damages caused to his career by Nazi Germany. Thevolume also includes two reports written by Busemann on his own work, aletter from Busemann to Victor Pambuccian, written in 1989, and a shorttext on memories of Busemann written by Michael Waterman.The first of the five essays included in this volume is written by AnnetteA’Campo-Neuen and the author of this introduction. It consists in acommentary to the paper Krümmungseigenschaften konvexer Fl¨achen (Curvatureproperties of convex surfaces), written in 1935, by Busemann andWilliam Feller. This paper is important for several reasons: First, it is oneof the first papers of Busemann on convexity theory, a topic that becameone of Busemann’s main subjects of interest during more than 35 years.The paper is also important because we see in it Busemann’s first attemptsto do geometry with a minimum amount of differentiability, which becamealso one of Busemann’s lifelong mathematical themes. The questions addressedin this paper belong to the main subject of Busemann’s monographConvex surfaces, published in 1958.The second essay is written by Valerii Berestovskii, and it is a surveyof Busemann’s work on convexity theory. This work is included by Beappearedin the same period on the same subject, starting with the worksof Bonnesen-Fenchel, Steinitz-Rademacher, and especially, the later worksby A. D. Alexandrov and his School. Particular attention is given inthis article to the problems that were posed by Busemann and his collaborators,and which became the foundations of a large portion of theresearch conducted on convexity theory. They concern area and volume, theBrunn-Minkowski theory and isoperimetric problems in Minkowski spaces.Most of all, Berestovskii discusses the famous Busemann-Petty problems,several of which remain until today among the most difficult and interestingopen problems in convexity theory and integral geometry. Berestovskiiprovides detailed comments on these problems, their (partial) solutions andtheir ramifications, with a review of the modern literature on this subject.Let us stress on the fact that like for the work on metric geometry that iscontained in Volume I of the present Selected Works, there are strong connectionsbetween Busemann’s work on convexity and that of A. D. Alexandrovand the school he founded on the same subject. As a matter of fact,Busemann’s monograph Convex surfaces (1958) was initially conceived asa book on the work of the Russian School on convexity. The correspondencekept at the Elmer Holmes Bobst Library includes a copy of a letter toBusemann, dated July 23, 1956 from Interscience Publishers, signed by E.S. Proskauer, Editor-in-Chief, with a copy to Courant, Stocker, and Bers,saying:Dear Dr. Busemann,It is with very special pleasure that we learn from Dr. Stockerof your willingness to present in our Tracts series an account ofthe important recent development of the work of certain Russiangeometers.This book was translated into Russian in 1962, and it is reviewed in detailin Berestovskii’s essay in the present volume.The third essay is written by Dmitri Burago, and it concerns more specificallyBusemann’s problems on convexity and integral geometry. As theauthor explains, several of these problems have many facets, and they constitutedthe basis of his joint research with Sergei Ivanov. In particular,Burago and Ivanov gave an affirmative answer to one aspect of Busemann’sproblem asking whether flats are area-minimizing in normed spaces. Theyconsidered the case of two-flats, both for the Hausdorff measure and for theHolmes-Thompson volume. A detailed and interesting review of differentpoints of view on this problem is presented in Burago’s essay.The fourth and the fifth essay are written by the author of this introduction.They concern two topics that accompanied Busemann duringmany years, and which are related to Hilbert’s Problem IV, asking for acharacterization of the geometries on subsets of Euclidean n-space or ndimensionalreal projective space for which the (Euclidean, or projective)lines are geodesics. The problem also asks for the study of these individualmetrics. The two essays concern respectively Busemann’s work onMinkowski geometries (the geometries of finite-dimensional normed spaces)and Hilbert geometries. These two classes of metrics satisfy Hilbert’s requirementsand are the two classes that are mentioned by Hilbert in hisstatement of Problem IV. Busemann’s results on these two topics are spreadin several of his books and papers, and we thought that it would be usefulto collect in these two papers his work on the subject. The characterizationof Minkowski geometries attracted Busemann since his early works, and, asa matter of fact, was the subject of his doctoral dissertation. RegardingHilbert geometry, the works of Busemann and his collaborators constitute,with two exceptions (a paper by Birkhoff and another one by Samelson)the only work done on this topic during the hundred years that followedHilbert’s discovery of that geometry. Hilbert geometry, since a couple ofdecades, became an important subject of research, in particular as part ofthe study of projective structures on manifolds. Busemann’s work on thissubject remains poorly known, with several of his results rediscovered withan ignorance of his original results. This is one of the main reasons that ledus to writing this survey.Let us mention that Busemann was the main promoter of Hilbert’s ProblemIV, and his work acted as the main catalyst to the solution of thatproblem that was given by A. V. Pogorelov in 1973. Let us quote the latter,from the introduction to his monograph Hilbert’s fourth problem:The occasion for the present investigation is a remarkable idea dueto Herbert Busemann, which I learned from his report to the InternationalCongress of Mathematicians at Moscow in 1966. Busemanngave an extremely simple and very general method of constructingDesarguesian metrics by using a nonnegative completelyadditive set function on the set of planes and defining the lengthof a segment as the value of the function of the set of planes intersectingthe segment.Busemann’s construction mentioned by Pogorelov is based on an integralformula for distances defined as a measure on the set of planes, contained inearlier work of Blaschke, which in turn is based on an integral formula dueto Crofton. The formula was promoted by Busemann in order to constructmetrics satisfying Hilbert’s problem IV.Let us also mention that Hilbert’s Fourth Problem has several interpretations,and in some respect it is still an open problem (in particular for whatconcerns non-symmetric metrics).The collection of papers by Busemann on convexity and on Hilbert’s ProblemIV that are contained in the present volume include several survey articleswhich the reader can take advantage of. We mention in particular hispaper with Feller, Regularity properties of a certain class of surfaces (1945),his paper with Petty, Problems on convex bodies (1956), his surveys OnHilbert’s fourth problem (1964), Desarguesian spaces (1976), Locally Desarguesianspaces, with B. B. Phadke (1980) and his book review on Pogorelov’sHilbert’s fourth problem (1981).The last section of Busemann’s papers which are reproduced in the presentvolume contains 6 papers on various topics, some of them on specializedsubjects, like his paper with Feller, Zur Differentiation der LebesgueschenIntegrale, (1934) and others on elementary topics, like his survey of non-Euclidean geometry (1950) or his paper Two applications of geometry (1967).The reason of the inclusion of a paper in this last section is that it was notobvious to which of the preceding sections it belongs (some of them couldbelong to more than one section).I would like to make a few comments on the paper The Role of Geometryfor the Mathematics Student in which Busemann reports on geometry incollege curricula, in 1960. His first observation is that “the standard course‘Analytic geometry and calculus’ originated from the waning interest in geometryand almost always proceeds at the expense of geometry.” Differentialgeometry is taught at a reasonable number of colleges, but, Busemann says,“there are influential mathematicians doubting the value of any of the remainingcourses.” The list of remaining courses is: analytic geometry (whichis not taught in the “Analytic geometry and calculus” course), projective geometry,foundations of geometry, non-Euclidean geometry, higher geometryand modern Euclidean geometry (triangle and circle). The usual objectionfor including these subjects in the curricula is that “the topic is no longer anactive subject of research.” Busemann’s defense for a subject like projectivegeometry is that “the subject is beautiful, it required major efforts of someof the best mathematicians during and after the last century to understandits structure thoroughly, it still gives a deep insight into geometry . . . Butmost of all, do we have the right to completely disrupt historical continuitywhenever a subject moves out of the focus of contemporary interests?Do we really expect or agree that our present mathematical efforts will bealtogether junked (at least from courses) as soon as the interests change?”Busemann would have certainly be pleased to know that todayprojective geometry, the subject of a book he wrote with Kelly in 1953,is today, like his metric geometry, at the forefront of research, in topologyand geometry. The same holds for non-Euclidean geometry, after the vasthorizons opened up by Thurston. In his last book, written with Phadke,Spaces with distinguished geodesics (1987), Busemann writes (p. 111) aboutthe importance of using metric methods in the study of Teichm¨uller spaces.This is also a topic that is extremely active today.The return to basic geometry is, contrary to what one may think, a difficultmatter, and discovering something new, using basic geometry, requires adeep understanding of mathematics. One is reminded here of A. D. Alexandrov’sSlogan: “Retreat to Euclid,”3 and of Ren´e Thom’s apology of returningto elementary geometry: “For Euclidean geometry, I keep a soft spotwhich my colleagues and my fellow mathematicians do not forgive me.”One is also reminded, of course, of Bill Thurston’s approach to mathematics:everything is based on elementary geometry.Busemann worked on fundamental questions, independently of the fashionof the day; this is probably the seal of a great mathematician.
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