Discrete Curvatures and Convex Polytopes

Abstract

Abstract. We study Forman–Ricci and effective resistance curvatures on the skeleta of convex polytopes. Our guiding questions are, How frequently do polytopal graphs exhibit everywhere-positive curvature, and what structural constraints does positivity impose? For Forman–Ricci curvature, we derive an exact identity for the average edge curvature in terms of flag [Formula: see text]-numbers and establish the existence of infinite families of Forman–Ricci-positive polytopes in every fixed dimension [Formula: see text]. We prove finiteness results in low dimension: there are only finitely many Forman–Ricci-positive 3- and 4-polytopes; for [Formula: see text] we show finiteness in the simplicial case and conjecture its extension to 5-polytopes more generally. For the resistance curvature [Formula: see text] we establish the existence of infinite families of resistance-positive polyhedral graphs for all [Formula: see text], and we provide a quantitative lower bound for [Formula: see text] in a simple 3-polytope in terms of the lengths of the three 2-faces incident to [Formula: see text]. This bound leads to constructions of non-vertex transitive, resistance-positive 3-polytopes via [Formula: see text]-operations, and a degree-based obstruction showing that if each neighbor of [Formula: see text] has degree at most [Formula: see text], then [Formula: see text]. Our results suggest that positive curvature on polytopal skeleta is rare and constrained.

The authors' abstract, as published at the source. SIAM Journal on Discrete Mathematics, 2026 · DOI ↗