Abstract
We show that the Banach–Mazur distance from any centrally symmetric convex body in ℝn to the n-dimensional cube is at most
which improves previously known estimates for “small” n≥3. (For large n, asymptotically better bounds are known; in the asymmetric case, exact bounds are known.) The proof of our estimate uses an idea of Lassak and the existence of two nearly orthogonal contact points in John’s decomposition of the identity. Our estimate on such contact points is closely connected to a well-known estimate of Gerzon on equiangular systems of lines.
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The author was partially supported by an Alexander Graham Bell Canada Graduate Scholarship and by a Vanier Canada Graduate Scholarship, both from the Natural Sciences and Engineering Research Council of Canada. The author also thanks his supervisor Alexander Litvak for his support and advice and Márton Naszódi for his valuable comments on an early draft of this note.
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Taschuk, S. The Banach–Mazur Distance to the Cube in Low Dimensions. Discrete Comput Geom 46, 175–183 (2011). https://doi.org/10.1007/s00454-010-9251-6
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DOI: https://doi.org/10.1007/s00454-010-9251-6