Hostname: page-component-78c5997874-4rdpn Total loading time: 0 Render date: 2024-11-10T22:07:28.658Z Has data issue: false hasContentIssue false
  • English
  • Français

Volumes of projection bodies

Part of: General convexity

Published online by Cambridge University Press:  26 February 2010

Noah Samuel Brannen
Noah Samuel Brannen
Affiliation:
2-36-6 Tama-cho, Fuchu-shi, Tokyo 183, Japan.
Get access

Abstract

C. M. Petty has conjectured the minimum value for a certain affine-invariant functional denned on the class of convex bodies. We give sharp bounds for this functional on a certain subclass of convex bodies, and we give a counterexample to an upper bound proposed by R. Schneider for the class of centrally symmetric convex bodies. We conjecture that the simplex provides the maximum on the class of all convex bodies, while the largest centrally symmetric subset of a simplex gives a sharp upper bound on the class of all centrally symmetric convex bodies.

MSC classification

Secondary: 52A40: Inequalities and extremum problems
Type
Research Article
Information
Mathematika , Volume 43 , Issue 2 , December 1996 , pp. 255 - 264
Copyright
Copyright © University College London 1996

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.Chakerian, G. D. and Logothetti, D.. Cube slices, pictorial triangles, and probability. Mathematics Magazine, 64 (1991), 219–241.CrossRefGoogle Scholar
2.Fáry, I. and Rédei, L.. Der zentralsymmetrische Kern und die zentralsymmetrische Hülle von konvexen Kürpern. Math. Ann., 122 (1950), 205–220.CrossRefGoogle Scholar
3.Gardner, R. J.. Geometric Tomography (Cambridge University Press, Cambridge, 1995).Google Scholar
4.Macbeath, A. M.. A compactness theorem for affine equivalence-classes of convex regions. Canad. J. Math., 3 (1951), 54–61.Google Scholar
5.McMullen, P.. Volumes of projections of unit cubes. Bull. London Math. Soc., 16 (1984), 278–280.CrossRefGoogle Scholar
6.Petty, C. M.. Projection bodies. In Proc. Colloq. Convexity, Copenhagen, 1965, (Københavns Univ. Mat. Inst., 1965) 234–241.Google Scholar
7.Petty, C. M.. Isoperimetric problems. In Proc. Conf. Convexity Combinat. Geom. (Univ. Oklahoma, 1971), 26–41.Google Scholar
8.Rogers, C. A. and Shephard, G. C.. The difference body of a convex body. Arch. Math., 8 (1957), 220 233.CrossRefGoogle Scholar
9.Schneider, R.. Random hyperplanes meeting a convex body. Z. Wahrscheinlichkeitsth. verw. Geb., 61 (1982), 379 387.CrossRefGoogle Scholar
10.Schneider, R.. Convex Bodies: The Brunn-Minkowski Theory (Cambridge University Press, Cambridge, 1993).CrossRefGoogle Scholar