Hostname: page-component-78c5997874-v9fdk Total loading time: 0 Render date: 2024-11-11T01:42:53.504Z Has data issue: false hasContentIssue false
  • English
  • Français

Centrally symmetric convex bodies

Part of: General convexity

Published online by Cambridge University Press:  26 February 2010

Paul R. Goodey
Paul R. Goodey
Affiliation:
Mathematics Department, University of Oklahoma, Norman, OK 73019, U.S.A.
Get access

Abstract

The principal objective of this work is to investigate various classes of centrally symmetric convex sets. These classes range from the zonoids at one extreme to the class of all centrally symmetric bodies at the other. The defining properties of these classes involve inequalities between mixed volumes. Various other characterizations will be found in response to a number of questions in a recent survey article by Rolf Schneider and Wolfgang Weil. Some of these are concerned with measures on a Grassmannian manifold while others relate to the intermediate surface area measures of convex bodies. We shall also show these classes are characterized by certain extremal geometric inequalities. The work concludes with a brief discussion of related results concerned with generalized zonoids.

MSC classification

Secondary: 52A40: Inequalities and extremum problems
Type
Research Article
Information
Mathematika , Volume 31 , Issue 2 , December 1984 , pp. 305 - 322
Copyright
Copyright © University College London 1984

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1.Aleksandrov, A. D.. Zur Theorie der gemischten Volumina von konvexen Korpern I, II, III (Russian with German summary). Mat. Sbornik N.S., 2 (1937), 947–972, 1205-1238; N.S., 3 (1938), 27-46.Google Scholar
2.Betke, U. and Goodey, P. R.. Continuous translation invariant valuations on convex bodies. Abh. Math. Sem. Hamburg, 54 (1984), 95–105.CrossRefGoogle Scholar
3.Bonnesen, T. and Fenchel, W.. Theorie der konvexen Körper (Springer, Berlin, 1934).Google Scholar
4.Choquet, G.. Lectures on analysis (Benjamin, Reading, Mass., 1969).Google Scholar
5.Fenchel, W. and Jessen, B.. Mengenfunktionen und konvexe Korper. Danske Vid. Selsk. Mat.-Fys. Medd., 16 (1938), 1–31.Google Scholar
6.Goodey, P. R.. Centrally symmetric convex sets and mixed volumes. Mathematika, 24 (1977), 193–198.CrossRefGoogle Scholar
7.Goodey, P. R. and Weil, W.. A uniqueness theorem for stationary Poisson networks. To appear in J. Appl. Prob.Google Scholar
8.Horvath, J.. Topological vector spaces and distributions Vol. 1 (Addison-Wesley, Reading, Mass., 1966).Google Scholar
9.Matheron, G.. Random sets and integral geometry (Wiley, New York, 1975).Google Scholar
10.Schneider, R.. Zu einem Problem von Shephard uber die Projektionen konvexer Korper. Math. Zeit., 101 (1967), 71–82.CrossRefGoogle Scholar
11.Schneider, R.. Additive Transformationen konvexer Korper. Geometriae Dedicata, 3 (1974), 221–228.CrossRefGoogle Scholar
12.Schneider, R. and Weil, W.. Zonoids and related topics. Convexity and its applications (Eds. Gruber, P. and Wills, J. M.) (Birkhauser, Basel, 1983).Google Scholar
13.Treves, F.. Topological vector spaces, distributions and kernels (Academic Press, New York, 1967).Google Scholar
14.Weil, W.. Centrally symmetric convex bodies and distributions. Israel Jour. Math., 24 (1976), 352–367.CrossRefGoogle Scholar
15.Weil, W.. Centrally symmetric convex bodies and distributions II. Israel Jour. Math., 32 (1979), 173–182.CrossRefGoogle Scholar
16.Weil, W.. On surface area measures of convex bodies. Geometriae Dedicata, 9 (1980), 299–306.CrossRefGoogle Scholar
17.Weil, W.. Zonoide und verwandte Klassen konvexer Korper. Monatshejte Math., 94 (1982), 73–84.CrossRefGoogle Scholar