  We provide an affirmative answer to a problem posed by Barvinok and Veomett,
showing that in general an n-dimensional convex body cannot be approximated by
a projection of a section of a simplex of a sub-exponential dimension.
Moreover, we establish a lower bound of the Banach-Mazur distance between
n-dimensional projections of sections of an N-dimensional simplex and a certain
convex symmetric body, which is sharp up to a logarithmic factor for all N>n.
