A geometric lower bound on the extension complexity of polytopes based on the f-vector

A linear extension of a polytope Q is any polytope which can be mapped onto Q via an affine transformation. The extension complexity of a polytope is the minimum number of facets of any linear extension of this polytope. In general, computing the extension complexity of a given polytope is difficult. The extension complexity is also equal to the nonnegative rank of any slack matrix of the polytope. In this paper, we introduce a new geometric lower bound on the extension complexity of a polytope, i.e., which relies only on the knowledge of some of its geometric characteristics. It is based on the monotone behaviour of the f-vector of polytopes under affine maps. We present numerical results showing that this bound can improve upon existing geometric lower bounds, and provide a generalization of this lower bound for the nonnegative rank of any matrix. (C) 2020 Elsevier B.V. All rights reserved.