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Estimating sizes of a convex body by successive diameters and widths

Part of: General convexity

Published online by Cambridge University Press:  26 February 2010

U. Betke  and
M. Henk
U. Betke
Affiliation:
Mathematisches Institut, Universität Siegen, Hōlderlinstrasse 3, D-W-5900, Siegen, Germany.
M. Henk
Affiliation:
Mathematisches Institut, Universität Siegen, Hölderlinstrasse 3, D-W-5900, Siegen, Germany.
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Abstract

The second theorem of Minkowski establishes a relation between the successive minima and the volume of a 0-symmetric convex body. Here we show corresponding inequalities for arbitrary convex bodies, where the successive minima are replaced by certain successive diameters and successive widths.

We further give some applications of these results to successive radii, intrinsic volumes and the lattice point enumerator of a convex body.

MSC classification

Secondary: 52A40: Inequalities and extremum problems
Type
Research Article
Information
Mathematika , Volume 39 , Issue 2 , December 1992 , pp. 247 - 257
Copyright
Copyright © University College London 1992

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References

BHe1.Betke, U. and Henk, M.. A generalization of Steinhagen's theorem. To appear in Abh. Math. Sent. Univ. Hamburg.Google Scholar
BHe2.Betke, U. and Henk, M.. Metrical properties of successive minima. To appear in Abh. Math. Sent. Univ. Hamburg.Google Scholar
BW.Betke, U. and Wills, J. M.. Stetige und diskrete Funktionale konvexer Körper. Contributions to Geometry (Proc. Geom. Sympos. Siegen, 1978), Ed. by Tölke, J. and Wills, J. M. (Birkhäuser, Basel, 1979), 226–237.CrossRefGoogle Scholar
BokHW.Bokowski, J.Hadwiger, H. and Wills, J. M.. Eine Ungleichung zwischen Volumen, Oberfläche und Gitterpunktanzahl konvexer Körper im n-dimensionalen euklidischen Raum. Math. Z., 127 (1972), 363–364.CrossRefGoogle Scholar
BoF.Bonnesen, T. and Fenchel, W.. Theorie der konvexen Körper (Springer, Berlin, 1934).Google Scholar
D.Davis, C.. In Convexity (Klee, V., ed.) Section: Unsolved problems. Amer. Math. Soc Proc. Symp. Pure Math., 13 (1963), 496.Google Scholar
GrL.Gruber, P. M. and Lekkerkerker, C. G.. Geometry of Numbers (North-Holland, Amsterdam, 1987).Google Scholar
GK.Gritzmann, P. and Klee, V.. Inner and outer j-radii of convex bodies in finite-dimensional normed spaces. Dis. and Comp. Geometry, 7 (1992), 255–281.CrossRefGoogle Scholar
H.Hadwiger, H.. Vorlesungen über Ihalt, Oberfläche und Isoperimetrie (Springer, Berlin, 1957).CrossRefGoogle Scholar
He1.Henk, M.. Ungleichungen für sukzessive Minima und verallgemeinerte In- und Umkugelradien konvexer Körper. Dissertation (Universitat Siegen, 1991).Google Scholar
He2.Henk, M.. A generalization of Jung's theorem. Geometriae Dedicata, 42 (1992), 235–240.CrossRefGoogle Scholar
He3.Henk, M.. Inequalities between successive minima and intrinsic volumes of a convex body. Mh. Math., 110 (1990), 279–282.CrossRefGoogle Scholar
Ku.Kubota, T.. Eine Ungleichheitsbeziehung über Eilinien und Eiflächen. Sci. Rep. Tôhoku Univ., 12 (1923), 45–65.Google Scholar
M.McMullen, P.. Non-linear angle-sum relations for polyhedral cones and polytopes. Math. Proc Camb. Phil. Soc., 78 (1975), 247–261.CrossRefGoogle Scholar
MS.McMullen, P. and Schneider, R.. Valuations on convex bodies. Contributions to Geometry (Proc. Geom. Sympos. Siegen, 1978), Ed. by Tolke, J. and Wills, J. M. (Birkhauser, Basel, 1979), 170–247.Google Scholar
Mi.Minkowski, H.. Geometrie der Zahlen (Teubner, Leipzig, 1910).Google Scholar
P.Perelmann, G. Y.. On the fk-radii of a convex body (in Russian). Sib. Mat. Zh., 28 (1987), 185–186.Google Scholar
S.Schneider, R.. Crofton's formula generalized to projected thick sections. Rend. Circ. Mat. Palermo (2), 30 (1981), 157–160.CrossRefGoogle Scholar
St.Steinhagen, P.. Uber die gröβte Kugel in einer konvexen Punktmenge. Abt. math. Seminar Hamburg 1, 1 (1922), 15–26.CrossRefGoogle Scholar
W.Wills, J. M.. Zur Gitterpunktanzahl konvexer Mengen. Elemente der Mathematik, 28 (1972), 57–63.Google Scholar
Z.Zindler, K.. Über konvexe Gebilde. I., II., III., MH. Math. Phys., 30 (1920), 87–102.CrossRefGoogle Scholar