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On Maximal $k$-Sections and Related Common Transversals of Convex Bodies

Published online by Cambridge University Press:  20 November 2018

Endre Makai Jr.  and
Horst Martini
Endre Makai Jr.
Affiliation:
E. Makai, Jr. Alfréd Rényi Mathematical Institute Hungarian Academy of Sciences Pf. 127 H-1364 Budapest Hungary, Email: [email protected]
Horst Martini
Affiliation:
H. Martini University of Technology Chemnitz Faculty of Mathematics D-09107 Chemnitz Germany, Email: [email protected]
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Abstract

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Generalizing results from $[\text{MM}1]$ referring to the intersection body $\text{IK}$ and the cross-section body $CK$ of a convex body $K\,\,\subset \,\,{{\mathbb{R}}^{d}}$, $d\,\ge \,2$, we prove theorems about maximal $k$-sections of convex bodies, $k\,\in \,\{1,\,\ldots ,\,d\,-\,1\}$, and, simultaneously, statements about common maximal $(d\,-\,1)-$ and 1-transversals of families of convex bodies.

Keywords

52A2055Mxx
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 47 , Issue 2 , 01 June 2004 , pp. 246 - 256
Copyright
Copyright © Canadian Mathematical Society 2004

References

[BF] Bonnesen, T. and Fenchel, W., Theorie der konvexen Körper. Springer, Berlin 1934 (korr. Nachdruck: 1974); Engl. transl.: Theory of convex bodies, BCS Associates, Moscow, ID, 1987.Google Scholar
[Bo] Bourbaki, N., Elements of Mathematics: General Topology, 1. Addison-Wesley, Reading, Massachusetts, 1966.Google Scholar
[Fe] Federer, H., Geometric measure theory. Grundlehren Math. Wiss., 153, Springer-Verlag, New York, 1969.Google Scholar
[Ga] Gardner, R. J., Geometric Tomography. Cambridge University Press, Cambridge, 1995.Google Scholar
[Gr] Gruber, P. M., Baire categories in convexity. In: Handbook of Convex Geometry (Eds. P. M. Gruber and J. M.Wills), North-Holland, Amsterdam, 1993, pp. 1327–1346.Google Scholar
[Grü] Grünbaum, B., On some properties of convex sets. Coll. Math. 8 (1961), 39–42.Google Scholar
[Ha1] Hammer, P. C., Convex bodies associated with a convex body. Proc. Amer.Math. Soc. 2 (1951), 781–793.Google Scholar
[Ha2] Hammer, P. C., Convex curves of constant Minkowski breadth. In: Convexity, Proc. Symp. Pure Math. 7, Ed. V. Klee, Amer. Math. Soc., Providence, RI, 1963, pp. 291–304.Google Scholar
[HW] Hurewicz, W. and Wallman, H., Dimension Theory. Princeton University Press, Princeton, NJ, 1941.Google Scholar
[Lu] Lutwak, E., Intersection bodies and dual mixed volumes. Adv. Math. 71 (1988), 232–261.Google Scholar
[MM1] Makai, E. Jr. and Martini, H.. On bodies associated with a given convex body. Canad. Math. Bull. 39 (1996), 448–459.Google Scholar
[MM2] Makai, E. Jr. and Martini, H., The cross-section body, plane sections of convex bodies and approximation of convex bodies, I. Geom. Dedicata 63 (1996), 267–296.Google Scholar
[MM3] Makai, E. Jr. and Martini, H., The cross-section body, plane sections of convex bodies and approximation of convex bodies, II. Geom. Dedicata 70 (1998), 283–303.Google Scholar
[MMO´ ] Makai, E. Jr., Martini, H. and O´dor, T., Maximal sections and centrally symmetric bodies. Mathematika 47 (2000), 19–30.Google Scholar
[MVŽ] Makai, E., Vre´cica, S. and Živaljevi´c, R., Plane sections of convex bodies of maximal volume. Discrete Comput. Geom. 25 (2001), 33–49.Google Scholar
[Ma1] Martini, H., On inner quermasses of convex bodies. Arch. Math. 52 (1989), 402–406.Google Scholar
[Ma2] Martini, H., Extremal equalities for cross-sectional measures of convex bodies. Proc. 3rd Geometry Congress, Aristoteles University Press, Thessaloniki, 1992, pp. 285–296.Google Scholar
[PC] Petty, C. M. and Crotty, J. M., Characterizations of spherical neighbourhoods. Canad. J. Math. 22 (1970), 431–435.Google Scholar
[Sch] Schneider, R., Convex Bodies: The Brunn-Minkowski Theory. Cambridge University Press, Cambridge, 1993.Google Scholar
[Sz] Szabó, Z. I., A short topological proof for the symmetry of 2 point homogenous spaces. Invent. Math. 106 (1991), 61–64.Google Scholar
[We] Weil, W., Ein Approximationssatz für konvexe körper. Manuscr. Math. 8 (1973), 335–362.Google Scholar
[Wh] Whittlesey, E. F., Fixed points and antipodal points. Amer. Math.Monthly 70 (1963), 807–821.Google Scholar