Some remarks on the geometry of convex sets

  • Keith Ball
Conference paper
Part of the Lecture Notes in Mathematics book series (LNM, volume 1317)


We prove a strengthening of Santalo's inequality for the unit balls of normed spaces with 1-unconditional bases and observe that all central sections of the unit cube in Rn (for n ≥ 10) have smaller volume than those of the Euclidean ball of volume 1.


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  1. 1.
    K.M. Ball, Cube slicing in R n, Proc. Amer. Math. Soc. 97 3 (1986), 465–473.MathSciNetMATHGoogle Scholar
  2. 2.
    K.M. Ball, Logarithmically concave functions and sections of convex sets, Studia Math, to appear.Google Scholar
  3. 3.
    K.M. Ball, Isometric problems and sections of convex sets, Dissertation, University of Cambridge, England (1986).Google Scholar
  4. 4.
    Herm Jan Brascamp and Elliott H. Lieb, On extensions of the Brunn-Minkowski and Prékopa-Leindler theorems, including inequalities for log. concave functions, and with an application to the diffusion equation, J.F.A. 4 (1976), 366–389.MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    H. Busemann, Volumes in terms of concurrent cross-sections, Pacific J. Math. 3 (1953), 1–12.MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    H. Busemann and C.M. Petty, Problems on convex bodies, Math. Scand. 4 (1956), 88–94.MathSciNetMATHGoogle Scholar
  7. 7.
    D. Hensley, Slicing convex bodies — bounds for slice area in terms of the bodies' covariances, Proc. Amer. Math. Soc. 79 #4 (1980), 619–625.MathSciNetMATHGoogle Scholar
  8. 8.
    D.G. Larman and C.A. Rogers, The existence of a centrally symmetric convex body with central sections that are unexpectedly small, Mathematika 22 (1975), 164–175.MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    L. Leindler, On a certain converse of Hölder's inequality. II, Acta Sci. Math. 33 (1972), 217–223.MathSciNetMATHGoogle Scholar
  10. 10.
    A. Prékopa, Logarithmic concave measures with application to stochastic programming, Acta Sci. Math. 32 (1971), 301–316.MathSciNetMATHGoogle Scholar
  11. 11.
    Y. Rinott, On convexity of measures, Thesis, Weizmann Inst., Rehovot, Israel, 1973.MATHGoogle Scholar
  12. 12.
    J. Saint-Raymond, Sur le volume des corps convexes symétriques, Séminaire d'initiation à l'analyse, 20e Année, 1980/1, Exp. #11, eds. G. Choquet, M. Rogalski, J. Saint-Raymond (Publ. Math. Univ. Pierre et Marie Curie 46), Univ. Paris VI, Paris, 1981.Google Scholar
  13. 13.
    L.A. Santalo, Un invariante afin para los cuerpos convexos del espacio de n dimensiones, Portugal math. 8, Fasc. 4 (1949).Google Scholar

Copyright information

© Springer-Verlag 1988

Authors and Affiliations

  • Keith Ball
    • 1
  1. 1.Department of Pure Mathematics and Mathematical StatisticsCambridge UniversityCambridgeEngland

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