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The volume of the intersection of a convex body with its translates

Part of: General convexity

Published online by Cambridge University Press:  26 February 2010

M. Meyer ,
S. Reisner  and
M. Schmuckenschläger
M. Meyer
Affiliation:
Equipe d'Analyse, Université paris VI, F-75252-Paris, Cedex 05, France
S. Reisner
Affiliation:
Department of Mathematics, University of Haifa, Haifa, 31905, Israel
M. Schmuckenschläger
Affiliation:
Dept. Theoretical Mathematics, Weizmann Institute of Science, Rehovot, 76100, Israel
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Abstract

It is proved that for a symmetric convex body K in ℝn, if for some τ >0, |K ⋂ (x + τK) depends on ‖x‖K only, then K is an ellipsoid. As a part of the proof, smoothness properties of convolution bodies are studied.

MSC classification

Secondary: 52A38: Length, area, volume
Type
Research Article
Information
Mathematika , Volume 40 , Issue 2 , December 1993 , pp. 278 - 289
Copyright
Copyright © University College London 1993

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References

B.Bollobaś, B.. Area of the union of disks. Elem. d. Math., 23 (1968), 60–61.Google Scholar
B-L.Barany, I. and Larman, D. G.. Convex bodies, economic cap coverings, random polytopes. Mathematika, 35 (1988), 274–291.CrossRefGoogle Scholar
C-P.Capoyleas, V. and Pach, J.. On the perimeter of a point set in the plane. A. M. S. DIM A CS Series in Discrete Mathematics and Theoretical Computer Science 6 (1991), 67–76.CrossRefGoogle Scholar
Go.Gordon, Y.. Majorization of Gaussian processes and geometric applications. Probability Th. and Related Fields, 91 (1992), 251–267.CrossRefGoogle Scholar
Gr.Gromov, M.. Monotonicity of the volume of intersection of balls. ‘Geometrical Aspects of Functional Analysis’, Lecture Notes in Math., 1267 (Springer Verlag, Berlin, 1987), 1–4.Google Scholar
H.Hadwiger, H. H.. Ungeloste Pröbleme No. 11. Elem. d. Math., 11 (1956), 60–61.Google Scholar
K.Kneser, M.. Einige Bemerkungen üiber das Minkowskische Flëchenmass. Arch. Math., 6 (1955), 382–390.CrossRefGoogle Scholar
L.Leichtweiss, K.. Zur Affinoberfläche knovexer Köper. Manuscr. Math., 56 (1986), 429–464.CrossRefGoogle Scholar
M.Macbeath, A. M.. A theorem on non-homogeneous lattices. Annals of Math., 56 (1952), 269–293.CrossRefGoogle Scholar
M-R-1.Meyer, M. and Reisner, S.. Characterizations of ellipsoids by section-centroid location. Geom. Ded., 31 (1989), 345–355.CrossRefGoogle Scholar
M-R-2.Meyer, M. and Reisner, S.. A geometric property of the boundary of symmetric convex bodies and convexity of flotation surfaces. Geom. Ded., 37 (1991), 327–337.CrossRefGoogle Scholar
P.Petty, C. M.. Affine isoperimetric problems. ‘Discrete Geometry and Convexity’, Ann. N. Y. Acad. Sci., 440 (1985), 113–127.CrossRefGoogle Scholar
S-m.Schmuckenschläger, M.. The distribution function of the convolution square of a convex symmetric body in Rn, Israel J. Math., 78 (1992), 309–334.CrossRefGoogle Scholar
S-n.Schneider, R.. Boundary structure and curvature of convex bodies. Contributions to Geometry, Proc. Geom. Symp. Siegen, 1978, eds. Tolke, J. and Wills, J. (Birkhauser, 1979), 1359.Google Scholar
S-W.Schütt, C. and Werner, E.. Homothetic floating bodies. Geom. Ded. To appear.Google Scholar
TP.Thue Poulsen, E.. Problem 10. Math. Scand., 2 (1954), 346.Google Scholar