On approximations by projections of polytopes with few facets

TL;DR

This paper proves that in general, convex bodies cannot be well-approximated by projections of simplexes with few facets, establishing bounds on the approximation quality in high dimensions.

Contribution

It answers a longstanding open problem by showing limitations of approximating convex bodies with projections of simplexes, providing sharp bounds on the Banach-Mazur distance.

Findings

Convex bodies generally cannot be approximated by projections of simplexes of sub-exponential dimension.

Established a lower bound on the Banach-Mazur distance for such approximations.

The bounds are sharp up to a logarithmic factor for all N > n.

Abstract

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.