Israel Journal of Mathematics

, Volume 80, Issue 3, pp 337–349 | Cite as

The only convex body with extremal distance from the ball is the simplex

  • O. Palmon


We can extend the Banach-Mazur distance to be a distance between non-symmetric sets by allowing affine transformations instead of linear transformations. It was proved in [J] that for every convex bodyK we haved(K, D)≤n. It is proved that ifK is a convex body in ℝ n such thatd(K, D)=n, thenK is a simplex.


Convex Body Common Point Affine Transformation Subsidiary Condition Symmetric Convex 
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  1. [G]
    B. Grünbaum,Measures of symmetry for convex sets, Proceedings of Symposia in Pure Mathematics. ConvexityVII (1963), 233–270.Google Scholar
  2. [J]
    F. John,Extremum problems with inequalities as subsidiary conditions, Courant Anniversary Volume, New York, 1948, pp. 187–204.Google Scholar
  3. [MW]
    V.D. Milman and H. Wolfson,Minkowski spaces with extremal distance from the Euclidean space, Isr. J. Math.29 (1978), 113–131.MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Hebrew University 1992

Authors and Affiliations

  • O. Palmon
    • 1
  1. 1.Department of MathematicsTel Aviv UniversityIsrael

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