Hostname: page-component-586b7cd67f-l7hp2 Total loading time: 0 Render date: 2024-11-28T05:01:45.473Z Has data issue: false hasContentIssue false
  • English
  • Français

Determination of convex bodies by their brightness functions

Part of: General convexity

Published online by Cambridge University Press:  26 February 2010

R. J. Gardner  and
A. Volčič
R. J. Gardner
Affiliation:
Department of Mathematics, Western Washington University, Bellingham, WA 98225-9063, U.S.A.
A. Volčič
Affiliation:
Dipartimento di Scienze Matematiche, Universitá degli Studi di Trieste, 34100 Trieste, Italy.
Get access

Abstract

It is shown that a convex body is determined uniquely among all convex bodies by the volumes of its projections onto all hyperplanes through the origin if and only if it is a parallelotope.

MSC classification

Secondary: 52A20: Convex sets in $n$ dimensions (including convex hypersurfaces)
Type
Research Article
Information
Mathematika , Volume 40 , Issue 2 , December 1993 , pp. 161 - 168
Copyright
Copyright © University College London 1993

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

A.Alexandrov, A. D.. On the theory of mixed volumes of convex bodies. II. New inequalities between mixed volumes and their applications (in Russian). Mat. Sb. N.S., 2 (1937), 1205–1238.Google Scholar
B.Blaschke, W.. Kreis und Kugel (Walter de Gruyter Co., Berlin, 1956).Google Scholar
BF.Bonnesen, T. and Fenchel, W.. Theory of Convex Bodies (BCS Associates, Moscow, Idaho, 1987).Google Scholar
BL.Bourgain, J. and Lindenstrauss, J.. Projection bodies. Geometric Aspects of Functional Analysis (Lindenstrauss, J. and Milman, V. D., eds.), Springer Lecture Notes, Vol. 1317 (Springer-Verlag, Berlin, 1988), 250–270.Google Scholar
F.Firey, W. J.. The brightness of convex bodies. Technical Report No. 19 (Oregon State University, Corvallis, Oregon, 1965).Google Scholar
GSW.Goodey, P., Schneider, R. and Weil, W.. Projection functions on higher rank Grassmannians. Geometric Aspects of Functional Analysis, edited by Lindenstrauss, J. and Milman, V. D.. To appear in Springer Lecture Notes.Google Scholar
G.Groemer, H.. Abschätzungen für die Anzahl der konvexen Körper, die einen konvexen Körper berühren. Monatsh. Math., 65 (1961), 74–81.CrossRefGoogle Scholar
Gr.Grünbaum, B.. On a conjecture of H. Hadwiger. Pacific J. Math., 11 (1961), 215–219.CrossRefGoogle Scholar
L.Lutwak, E.. Inequalities for mixed projection bodies. Trans. Amer. Math. Soc., To appear.Google Scholar
M.Martini, H.. Zur Bestimmung Konvexer Polytope durch die Inhalte ihrer Projektionen. Beitrdäe zur Algebra und Geometrie, 18 (1984), 75–85.Google Scholar
S.Schneider, R.. Boundary structure and curvature of convex bodies. In Contributions to Geometry (Tölke, J. and Wills, J. M., eds.) (Birkhäuser, Basel, 1979), 13–59.CrossRefGoogle Scholar
SW.Schneider, R. and Weil, W.. Zonoids and related topics. Convexity and its Applications (Gruber, P. M. and Wills, J. M., eds.) (Birkhäuser, Basel, 1983), 296–317.CrossRefGoogle Scholar
V.Vincensini, P.. Corps convexes. Séries linéaires. Domaines vectoriels. Mem. des Sciences Math., 94 (1938), 1–57.Google Scholar
Y.Yost, D.. Irreducible convex sets. Mathematika, 38 (1991), 134–155.CrossRefGoogle Scholar