On approximation by projections of polytopes with few facets |
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Authors: | Alexander E. Litvak Mark Rudelson Nicole Tomczak-Jaegermann |
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Affiliation: | 1. Depatment of Mathematical and Statistical Sciences, University of Alberta, Edmonton, Alberta, T6G 2G1, Canada 2. Department of Mathematics, University of Michigan East Hall, 530 Church Street, Ann Arbor, Michigan, 48109, USA 3. Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, Alberta, T6G 2G1, Canada
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Abstract: | We provide an affirmative answer to a problem posed by Barvinok and Veomett in [4], showing that in general an n-dimensional convex body cannot be approximated by a projection of a section of a simplex of subexponential dimension. Moreover, we prove that for all 1 ≤ n ≤ N there exists an n-dimensional convex body B such that for every n-dimensional convex body K obtained as a projection of a section of an N-dimensional simplex one has $$d(B,K) geqslant csqrt {frac{n}{{ln frac{{2Nln (2N)}}{n}}}} $$ , where d(·, ·) denotes the Banach-Mazur distance and c is an absolute positive constant. The result is sharp up to a logarithmic factor. |
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