On Compact Sets with a Certain Affine Invariant

A. S. Voynov

M. V. Lomonosov Moscow State University

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.

Keywords: compact set, symmetry, affine symmetry, convex body, ellipsoid, John ellipsoid, Lebesgue measure, second-order hypersurface.

UDC: 517

Received: 09.11.2009

DOI: 10.4213/mzm8618

 English version: Mathematical Notes, 2011, 90:1, 32–36

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