Convexity and Its Applications

1. Front Matter

   Pages 1-7

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2. Convexity and Optimization in Discrete Structures July 1982

   • Achim Bachem

   Pages 9-29

3. Isoperimetric inequalities

   • Catherine Bandle

   Pages 30-48

4. Convex Bodies of Constant Width

   • G. D. Chakerian, H. Groemer

   Pages 49-96

5. Algebraic Lattices

   • J. H. H. Chalk

   Pages 97-110

6. The Twenty-Seven Lines on the Cubic Surface

   • H. S. M. Coxeter

   Pages 111-119

7. Convexity Through the Ages

   • W. Fenchel

   Pages 120-130

8. Approximation of convex bodies

   • Peter M. Gruber

   Pages 131-162

9. Geometric convexity and differential geometry

   • Kurt Leichtweiss

   Pages 163-169

10. Valuations on convex bodies

    • Peter McMullen, Rolf Schneider

    Pages 170-247

11. Minimal and Closest Points Nonexpansive and Quasi-Nonexpansive Retractions in Real Banach Spaces

    • Pier Luigi Papini

    Pages 248-263

12. Ellipsoids

    • Clinton M. Petty

    Pages 264-276

13. Convexity in Banach spaces: some recent results

    • R. R. Phelps

    Pages 277-295

14. Zonoids and Related Topics

    • Rolf Schneider, Wolfgang Weil

    Pages 296-317

15. New Results in the Theory of Packing and Covering

    • G. Fejes Tóth

    Pages 318-359

16. Stereology: A Survey for Geometers

    • Wolfgang Weil

    Pages 360-412

17. Semi-Platonic Manifolds

    • J. M. Wills

    Pages 413-421

This collection of surveys consists in part of extensions of papers presented at the conferences on convexity at the Technische Universitat Wien (July 1981) and at the Universitat Siegen (July 1982) and in part of articles written at the invitation of the editors. This volume together with the earlier volume «Contributions to Geometry» edited by Tolke and Wills and published by Birkhauser in 1979 should give a fairly good account of many of the more important facets of convexity and its applications. Besides being an up to date reference work this volume can be used as an advanced treatise on convexity and related fields. We sincerely hope that it will inspire future research. Fenchel, in his paper, gives an historical account of convexity showing many important but not so well known facets. The articles of Papini and Phelps relate convexity to problems of functional analysis on nearest points, nonexpansive maps and the extremal structure of convex sets. A bridge to mathematical physics in the sense of Polya and Szego is provided by the survey of Bandle on isoperimetric inequalities, and Bachem's paper illustrates the importance of convexity for optimization. The contribution of Coxeter deals with a classical topic in geometry, the lines on the cubic surface whereas Leichtweiss shows the close connections between convexity and differential geometry. The exhaustive survey of Chalk on point lattices is related to algebraic number theory. A topic important for applications in biology, geology etc.

• optimization
• research
• science and technology
• sets

• Institut für Analysis, Technische Mathematik und Versicherungsmathematik, Technische Universität Wien, Wien, Austria

  Peter M. Gruber

• Lehrstuhl für Mathematik II, Universität Siegen, Siegen 21, Germany

  Jörg M. Wills

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