Hostname: page-component-cd9895bd7-8ctnn Total loading time: 0 Render date: 2024-12-27T21:24:55.606Z Has data issue: false hasContentIssue false
  • English
  • Français

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: makai@renyi.hu
H. Martini
Affiliation:
Techńische Universität Chemnitz, Fakultät für Mathematik, D-09107 Chemnitz, Germany. E-mail: martini@mathematik.tu-chemnitz.de
T. Ódor
Affiliation:
Alfréd Rényi Mathematical Institute, Hungarian Academy of Sciences, P.O.B. 127, H-1364 Budapest, Hungary. E-mail: odor@math.u-szeged.hu
Get access

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

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