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Maximal sections and centrally symmetric bodies

Part of: General convexity

Published online by Cambridge University Press:  26 February 2010

E. Makai Jr ,
H. Martini  and
T. Ódor
E. Makai Jr
Affiliation:
Alfréd Rényi Mathematical Institute, Hungarian Academy of Sciences, P.O.B. 127, H-1364 Budapest, Hungary. E-mail: [email protected]
H. Martini
Affiliation:
Techńische Universität Chemnitz, Fakultät für Mathematik, D-09107 Chemnitz, Germany. E-mail: [email protected]
T. Ódor
Affiliation:
Alfréd Rényi Mathematical Institute, Hungarian Academy of Sciences, P.O.B. 127, H-1364 Budapest, Hungary. E-mail: [email protected]
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Abstract

Let d≥2 and let K⊂ℝd be a convex body containing the origin 0 in its interior. For each direction ω, let the (d−l)-volume of the intersection of K and an arbitrary hyperplane with normal ω attain its maximum when the hyperplane contains 0. Then K is symmetric about 0. The proof uses a linear integro-differential operator on Sd−1, whose null-space needs to be, and will be determined.

MSC classification

Secondary: 52A40: Inequalities and extremum problems
Type
Research Article
Information
Mathematika , Volume 47 , Issue 1-2 , December 2000 , pp. 19 - 30
Copyright
Copyright © University College London 2000

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References

[A]Adams, R. A.. Sobolev Spaces, Academic Press Inc., New York-London (1975).Google Scholar
[Al]AlexandrofT, A. D.. Zur Theorie der gemischten Volumina von konvexen, Körpern II., Neue Ungleichungen zwischen den gemischten Volumina und ihre Anwendungen. Mat. Sb., 2 (1937), 1205–1238 (Russian with German summary).Google Scholar
[B]Blaschke, W.. Kreis und Kugel, 2-e durchges. verbess. Aufl. De Gruyter (Berlin 1956).CrossRefGoogle Scholar
[EMO]Erdélyi, A., Magnus, W., Oberhettinger, F., and Tricomi, F. G.. Higher Transcendental Functions II. McGraw-Hill (New-York-Toronto-London, 1953).Google Scholar
[F]Falconer, K. J.. Applications of a result on spherical integration to the theory of convex sets. Amer. Math. Monthly, 90 (1983), 690–693.CrossRefGoogle Scholar
[Ful]Funk, P.. Über Flächen mit lauter geschlossenen geodätischen Linien. Math. Ann., 74 (1913), 278–300.CrossRefGoogle Scholar
[Fu2]Funk, P.. Über eine geometrische Anwendung der Abelschen Integralgleichung. Math. Ann., 77 (1916), 129–135.CrossRefGoogle Scholar
[G]Gardner, R. J.. Geometric Tomography. Encyclopedia of Math, and its Appls., 58, Cambridge Univ. Press (Cambridge, 1995).Google Scholar
[Grl]Groemer, H.. Fourier series and spherical harmonics in convexity. In Handbook of Convex Geometry (Gruber, P. M. and Wills, J. M., eds.), North Holland (Amsterdam, 1993), 1259–1295.CrossRefGoogle Scholar
[Gr2]Groemer, H.. Geometric Applications of Fourier Series and Spherical Harmonics. Encyclopedia of Math, and its Appls., 61, Cambridge Univ. Press (Cambridge, 1996).CrossRefGoogle Scholar
[H]Hammer, P. C.. Diameters of convex bodies. Proc. Amer. Math. Soc, 5 (1954), 304–306.CrossRefGoogle Scholar
[Hel]Helgason, S.. The Radon Transform, Birkhäuser (Boston, Mass., 1980).CrossRefGoogle Scholar
[He2]Helgason, S.. Groups and Geometric Analysis. Integral Geometry, Invariant Differential Operators, and Spherical Functions. Academic Press, Inc. (Orlando, Fl., 1984).Google Scholar
[LP]Lifshitz, I. M., and Pogorelov, A. V.. On the determination of Fermi surfaces and electron velocities in metals by the oscillation of magnetic susceptibility (Russian). Dokl. Akad. Nauk SSSR, 96 (1954), 1143–1145.Google Scholar
[L]Lutwak, E.. Intersection bodies and dual mixed volumes. Advances Math., 71 (1988), 232–261.CrossRefGoogle Scholar
[M]Mackintosh, A. R.. The Fermi surface of metals. Scientific American, 209 (1963).CrossRefGoogle Scholar
[MM]Makai, E. Jr. and Martini, H.. On bodies associated with a given convex body. Canad. Math. Bull., 39 (1996), 448–459.CrossRefGoogle Scholar
[MM1]Makai, E. Jr. and Martini, H.. On an integro-differential transform on the sphere. Studia Sci. Math. Hungar. 38 (2001), 299–312.Google Scholar
[Mai]Martini, H.. Extremal equalities for cross-sectional measures of convex bodies. Proc. 3rd Geometry Congress, Aristoteles Univ. Thessaloniki (Thessaloniki 1992), 285–296.Google Scholar
[Ma2]Martini, H.. Cross-sectional measures. Intuitive Geom. (Szeged 1991), Coll. Math. Soc. J. Bolyai, 63, North Holland (Amsterdam-Oxford-New York, 1994), 269–310.Google Scholar
[Mu]Müller, C.. Spherical Harmonics. Lecture Notes in Math., 17, Springer (Berlin, 1966).CrossRefGoogle Scholar
[Pl]Petty, C. M.. On Minkowski Geometries. Ph.D. dissertation, Univ. South California (Los Angeles, 1952).Google Scholar
[P2]Petty, C. M.. Centroid surfaces. Pacific J. Math., 11 (1961), 1535–1547.CrossRefGoogle Scholar
[R]Radon, J.. Über die Bestimmung von Funktionen durch ihre Integralwerte langs gewisser Mannigfaltigkeiten. Ber. Verh. Sächs. Akad. Wiss. Leipzig, Math Nat. Kl., 69 (1917), 262–277.Google Scholar
[S]Sansone, G.. Orthogonal Functions (Rev. ed.). Wiley-Interscience (London-New York, 1959).Google Scholar
[SI]Schneider, R.. Zu einem Problem von Shephard über die Projektionen konvexer Korper. Math. Z., 101 (1967), 71–82.CrossRefGoogle Scholar
[S2]Schneider, R.. Functions on a sphere with vanishing integrals over certain subspheres. J. Math. Anal. Appl., 26 (1969), 381–384.CrossRefGoogle Scholar
[S3]Schneider, R.. Über eine Integralgleichung in der Theorie der konvexen Körper. Math. Nadchr., 44 (1970), 55–75.CrossRefGoogle Scholar
[S4]Schneider, R.. Convex Bodies: the Brunn-Minkowski Theory. Encyclopedia of Math. and its Appls., 44, Cambridge Univ. Press (Cambridge, 1993).CrossRefGoogle Scholar
[Se]Seeley, R. T.. Spherical harmonics. Amer. Math. Monthly, 73 (1966), 115–121.CrossRefGoogle Scholar
[Sh]Shoenberg, D.. The de Haas-van Alphen effect. In The Fermi Surface (ed. Harrison, W. A. and Web, M. B.), Wiley (New York, 1960), 74–83.Google Scholar
[Tl]Tricomi, F. G.. Vorlesungen uber Orthogonalreihen. Grundlehren Math. Wiss., 76, Springer-Verlag (Berlin-Göttingen-Heidelberg, 1955).CrossRefGoogle Scholar
[T2]Tricomi, F. G.. Integral Equations, Wiley-Interscience (New York-London, 1957).Google Scholar
[U]Ungar, P.. Freak theorem about functions on a sphere. J. London Math. Soc, 29 (1954), 100–103.CrossRefGoogle Scholar
[W]Whitney, H.. Geometric Integration Theory. Princeton Univ. Press (Princeton, N.J., 1957).CrossRefGoogle Scholar
[Z]Ziemer, W. P.. Weakly Differentiate Functions. Sobolev Spaces and Functions of Bounded Variation, Springer Verlag (New York, 1989).Google Scholar