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On compact sets with a certain affine invariant

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Abstract

We give a complete characterization of finite-dimensional compact sets with the following property: all of their images under affine operators are symmetric (that is, have symmetry planes of certain dimensions). We also study the noncompact case; namely, we reveal a class of unbounded closed sets with this property and conjecture that this class is complete.

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References

  1. L. Montejano, “Convex bodies with homothetic sections,” Bull. London Math. Soc. 23(4), 381–386 (1991).

    Article  MathSciNet  Google Scholar 

  2. H. Steinhaus, Sto zadań (PWN, Warszawa, 1958; Mir, Moscow, 1976).

    Google Scholar 

  3. G. R. Burton, “Congruent sections of a convex body,” Pacific J.Math. 81(2), 303–316 (1979).

    MathSciNet  Google Scholar 

  4. G. R. Burton, “Some characterisations of the ellipsoid,” Israel J. Math. 28(4), 339–349 (1977).

    Article  MathSciNet  Google Scholar 

  5. F. John, “Extremum problems with inequalities as subsidiary conditions,” in Studies and Essays Presented to R. Courant on his 60th Birthday (Interscience Publ., New York, 1948), pp. 187–204.

    Google Scholar 

  6. K. Chandrasekharan, Introduction to Analytic Number Theory (Berlin, Springer, 1969; Mir, Moscow, 1974).

    Google Scholar 

  7. P. S. Aleksandrov, A Course in Analytic Geometry and Linear Algebra (Mir, Moscow, 1979) [in Russian].

    MATH  Google Scholar 

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Authors and Affiliations

  1. Moscow State University, Moscow, Russia

    A. S. Voynov

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  1. A. S. Voynov
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Correspondence to A. S. Voynov.

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Original Russian Text © A. S. Voynov, 2011, published in Matematicheskie Zametki, 2011, Vol. 90, No. 1, pp. 34–39.

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Voynov, A.S. On compact sets with a certain affine invariant. Math Notes 90, 32 (2011). https://doi.org/10.1134/S0001434611070042

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  • DOI: https://doi.org/10.1134/S0001434611070042

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