Authors: Eran Nevo ¹; Guillermo Pineda-Villavicencio ²; Julien Ugon ²; David Yost ²

• 1 Einstein Institute of Mathematics
• 2 Centre for Informatics and Applied Optimization

this is an extended abstract of the full version. We study n-vertex d-dimensional polytopes with at most one nonsimplex facet with, say, d + s vertices, called almost simplicial polytopes. We provide tight lower and upper bounds for the face numbers of these polytopes as functions of d, n and s, thus generalizing the classical Lower Bound Theorem by Barnette and Upper Bound Theorem by McMullen, which treat the case s = 0. We characterize the minimizers and provide examples of maximizers, for any d.