Lower Bounds on Face Numbers of Polytopes with m Facets

2024/01/27 by Joshua G. Hinman, Hinman, Joshua · 1 citation
Mathematics · #52B05 #Advanced Combinatorial Mathematics #Combinatorics (math.CO) #FOS: Mathematics

paper · pdf · doi:10.48550/arxiv.2401.15361

openalex publication_date 2024/01/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let P be a convex d-polytope and 0 ≤ k ≤ d-1. In 2023, this author proved the following inequalities, resolving a question of Bárány: (f_k(P))/(f₀(P)) ≥ (1)/(2)\biggl[\lceil (d)/(2) \rceil \choose k + \lfloor (d)/(2) \rfloor \choose k\biggr], \fracf_k(P)f_d-1(P) ≥ (1)/(2)\biggl[\lceil (d)/(2) \rceil \choose d-k-1 + \lfloor (d)/(2) \rfloor \choose d-k-1\biggr]. We show that for any fixed d and k, these are the tightest possible linear bounds on f_k(P) in terms of f₀(P) or f_d-1(P). We then give a stronger bound on f_k(P) in terms of the Grassmann angle sum γ_k²(P). Finally, we prove an identity relating the face numbers of a polytope with the behavior of its facets under a fixed orthogonal projection of codimension two.

Cited by

• Approximation of the Euclidean ball by polytopes with a fixed number of k-faces

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