On approximation by projections of polytopes with few facets

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 sub-exponential 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) ≥ c √ n ln 2N ln(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. 2010 Subject Classification: Primary: 52A23, 52A27; Secondary: 52B55, 46B09.