Abstract
Many crucial results of the asymptotic theory of symmetric convex bodies were extended to the non-symmetric case in recent years. That led to the conjecture that for everyn-dimensional convex bodyK there exists a projectionP of rankk, proportional ton, such thatPK is almost symmetric. We prove that the conjecture does not hold. More precisely, we construct ann-dimensional convex bodyK such that for everyk >C√nlnn and every projectionP of rankk, the bodyPK is very far from being symmetric. In particular, our example shows that one cannot expect a formal argument extending the “symmetric” theory to the general case.
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This author holds a Lady Davis Fellowship.
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Gluskin, E.D., Litvak, A.E. & Tomczak-Jaegermann, N. An example of a convex body without symmetric projections. Isr. J. Math. 124, 267–277 (2001). https://doi.org/10.1007/BF02772622
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DOI: https://doi.org/10.1007/BF02772622