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Jeffrey M. Lemm, "Handbook of Convex Geometry, Volume B"

Preface

One aim of this Handbook is to survey convex geometry, its many ramifications

and its relations with other areas of mathematics. We believe that it will be a

useful tool for the expert. A second aim is to give a high level introduction to

most branches of convexity and its applications, showing the major ideas, methods

and results. We hope that because of this feature the Handbook will act as an

appetizer for future researchers in convex geometry. For them the many explicitly

or implicitly stated problems should turn out to be a valuable source of inspiration.

Third, the Handbook should be useful for mathematicians working in other areas

as well as for econometrists, computer scientists, crystallographers, phycisists and

engineers who are looking for geometric tools for their own work. In particular,

mathematicians specializing in optimization, functional analysis, number theory,

probability theory, the calculus of variations and all branches of geometry should

profit from the Handbook.

The famous treatise "Theorie der konvexen Korper" by Bonnesen and Fenchel

presented in 164 pages an almost complete picture of convexity as it appeared

around 1930. While a similarly comprehensive report today seems to be out of

reach, the Handbook deals with most of the more important topics of convexity

and its applications. By comparing the Handbook with the survey of Bonnesen and

Fenchel and with more recent collections of surveys of particular aspects of geo-

metric convexity such as the AMS volume edited by Klee (1963), the Copenhagen

Colloquium volume edited by Fenchel (1967), the two green Birkhauser volumes

edited by Tolke and Wills (1979) and Gruber and Wills (1983), respectively, and

the New York Academy volume edited by Goodman, Lutwak, Malkevitch and

Pollack (1985), the reader may see where progress was made in recent years.

During the planning stage of the Handbook, which started in 1986, we got gene-

rous help from many prominent convex geometers, in particular Peter McMullen,

Rolf Schneider and Geoffrey Shephard. The discussion of the list of contents and

of prospective authors turned out to be difficult. Both of us are obliged to the

authors who agreed to contribute to the Handbook. In the cooperation with them

we got much encouragement and the professional contacts furthered our good

personal relations with many of them. The manuscripts which we finally received

turned out to be much more diverse than we had anticipated. They clearly exhibit

the most different characters and scientific styles of the authors and this should

make the volume even more attractive.

There are several researchers in geometric convexity whom we invited to con-

tribute to the Handbook but who for personal. professional or other reasons -

regretfully - were not able to participate. The reader will also note that one area

or another is missing in the list of contents. Examples are elementary geometry

of normed planes, axiomatic convexity, and Choquet theory, but this should not

diminish the usefulness of the Handbook.

Some fields such as computational and algorithmic aspects or lattice point re-

sults are dealt with in several chapters. The subjects covered are organized in

five parts which in some sense reflect the fact that convexity is situated between

analysis, geometry and discrete mathematics. This organization clearly has some

disadvantages but we think that it will be helpful for the reader who wants to

orient himself.

In the editing of the Handbook we received much help, in particular from

Dr. M. Henk, Dr. J. Miiller, Professor S. Hildebrandt, Professor L. Payne and

Professor E Schnitzer. We are most grateful to Ms. S. Clarius and Ms. E. Rosta

who typed about 1000 letters to the authors, to colleagues whom we asked for

advice and to the Publishers. We most gratefully acknowledge the friendly coop-

erations and expert support of Dr. A. Sevenster and Mr. W. Maas from Elsevier

Science Publishers.

Finally we should like to express our sincere hope that the readers will appreciate

the great effort of so many prominent authors and will find the Handbook useful

for their scientific work.

Peter M. Gruber

Jorg M. Wills

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