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THE CONVEX INTERSECTION BODY OF A CONVEX BODY

Published online by Cambridge University Press:  10 March 2011

MATHIEU MEYER  and
SHLOMO REISNER
MATHIEU MEYER
Affiliation:
Université Paris-Est - Marne-la-Vallée, Laboratoire d'Analyse et de Mathématiques Appliquées (UMR 8050), Cité Descartes, 5 Bd Descartes, Champs-sur-Marne, 77454 Marne-la-Vallée cedex 2, France e-mail: mathieu.meyer@univ-mlv.fr
SHLOMO REISNER
Affiliation:
Department of Mathematics, University of Haifa, Haifa 31905, Israel e-mail: reisner@math.haifa.ac.il
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Abstract

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Let L be a convex body in n and z an interior point of L. We associate with L and z a new, convex and centrally symmetric, body CI(L, z). This generalizes the classical intersection bodyI(L, z) (whose radial function at u ∈ Sn−1 is the volume of the hyperplane section of L through z, orthogonal to u). CI(L, z) coincides with I(L, z) if and only if L is centrally symmetric about z. We study the properties of CI(L, z).

Keywords

52A20
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 53 , Issue 3 , September 2011 , pp. 523 - 534
Copyright
Copyright © Glasgow Mathematical Journal Trust 2011

References

REFERENCES

1.Ball, K., Logarithmically concave functions and sections of convex sets in n, Stud. Math. 88 (1988), 69–84.CrossRefGoogle Scholar
2.Brehm, U., Convex bodies with non-convex cross-section bodies, Mathematika 46 (1999), 127–129.CrossRefGoogle Scholar
3.Busemann, H., Volume in terms of concurrent cross-sections, Pac. J. Math. 3 (1953), 1–12.CrossRefGoogle Scholar
4.Campi, S. and Gronchi, P., Volume inequalities for sets associated with convex bodies, in Integral geometry and convexity (Grinberg, E. L., Li, S., Zhang, G. and Zhou, J., Editors) (World Sci. Publ., Hackensack, NJ, 2006), 1–15.Google Scholar
5.Campi, S. and Gronchi, P., On volume product inequalities for convex sets, Proc. Am. Math. Soc. 134 (2006), 2393–2402.CrossRefGoogle Scholar
6.Falconer, K. J., Applications of a result on spherical integration to the theory of convex sets, Am. Math. Monthly 90 (1983), 690–693.CrossRefGoogle Scholar
7.Fradelizi, M., Hyperplane sections of convex bodies in isotropic position, Beiträge Algebra Geom. 40 (1999), 163–183.Google Scholar
8.Funk, P., Über eine geometriche Anvendung der Abelschen Integralgleichung. Math. Ann. 77 (1916), 129–135.CrossRefGoogle Scholar
9.Gardner, R. J., Geometric tomography, in Encyclopedia of mathematics and its applications, Vol. 58, 2nd edn. (Cambridge University Press, Cambridge, UK, 2006).Google Scholar
10.Grünbaum, B., Measures of symmetry for convex sets, in Proceedings of symposia in pure mathematics, Vol. VII (American Mathematical Society, Providence, RI, 1963), 233–270.Google Scholar
11.Hensley, D., Slicing convex bodies—Bounds for slice area in terms of the body's covariance, Proc. Am. Math. Soc. 79 (1980), 619–625.Google Scholar
12.Lutwak, E., Intersection bodies and dual mixed volumes, Adv. Math. 71 (1988), 232–261.CrossRefGoogle Scholar
13.Makai, E. Jr., Martini, H. and Ódor, T., Maximal sections and centrally symmetric bodies, Mathematika 47 (2000), 19–30.CrossRefGoogle Scholar
14.Meyer, M., Maximal hyperplane sections of convex bodies, Mathematika 46 (1999), 131–136.CrossRefGoogle Scholar
15.Meyer, M. and Reisner, S., Characterizations of ellipsoids by section-centroid location, Geom. Dedicata 31 (1989), 345–355.CrossRefGoogle Scholar
16.Meyer, M. and Reisner, S., Shadow systems and volumes of polar convex bodies, Mathematika 53 (2006), 129–148 (2007).CrossRefGoogle Scholar
17.Meyer, M., Schütt, C. and Werner, E., A convex body whose centroid and Santaló point are far apart, arXiv:1001.0714v1 (preprint).Google Scholar
18.Meyer, M. and Werner, E., The Santaló-regions of a convex body, Trans. A.M.S. 350 (1998), 4569–4591.CrossRefGoogle Scholar
19.Milman, V. D. and Pajor, A., Isotropic position and inertia ellipsoids and zonoids of the unit ball of a normed n-dimensional space, in Geometric aspects of functional analysis (1987–88), Lecture Notes in Math., 1376 (Springer, Berlin, 1989), 64–104.CrossRefGoogle Scholar
20.Santaló, L. A., Un invariante afin para los cuerpos convexos del espacio de n dimensiones, Portugal Math. 8 (1949), 155–161.Google Scholar
21.Schneider, R., Convex bodies, the Brunn–Minkowski theory (Cambridge University Press, Cambridge, UK, 1993).CrossRefGoogle Scholar
22.Schütt, C., Floating body, illumination body, and polytopal approximation, in Convex geometric analysis (Berkeley, CA, 1996), 203–229; Math. Sci. Res. Inst. Publ. 34 (1999) (Cambridge University Press, Cambridge).Google Scholar
23.Shephard, G. C., Shadow systems of convex sets. Israel J. Math. 2 (1964), 229–236.CrossRefGoogle Scholar