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Extremal problems and isotropic positions of convex bodies

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Abstract

LetK be a convex body in ℝn and letW i (K),i=1, …,n−1 be its quermassintegrals. We study minimization problems of the form min{W i (TK)|T ∈ SL n } and show that bodies which appear as solutions of such problems satisfy isotropic conditions or even admit an isotropic characterization for appropriate measures. This shows that several well known positions of convex bodies which play an important role in the local theory may be described in terms of classical convexity as isotropic ones. We provide new applications of this point of view for the minimal mean width position.

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Authors and Affiliations

  1. Department of Mathematics, University of Crete, Iraklion, Greece

    A. A. Giannopoulos

  2. Department of Mathematics, Tel Aviv University, Tel Aviv, Israel

    V. D. Milman

Authors
  1. A. A. Giannopoulos
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  2. V. D. Milman
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Correspondence to A. A. Giannopoulos.

Additional information

Supported in part by a research grant of the University of Crete.

Supported in part by the Israel Science Foundation founded by the Academy of Sciences and Humanities.

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Giannopoulos, A.A., Milman, V.D. Extremal problems and isotropic positions of convex bodies. Isr. J. Math. 117, 29–60 (2000). https://doi.org/10.1007/BF02773562

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  • DOI: https://doi.org/10.1007/BF02773562

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