Discrete Curvatures and Convex Polytopes

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 f-numbers and establish the existence of infinite families of Forman--Ricci-positive polytopes in every fixed dimension dge 6. We prove finiteness results in low dimension: there are only finitely many Forman--Ricci-positive 3- and 4-polytopes; for d=5 we show finiteness in the simplicial case, and conjecture its extension to 5-polytopes more generally. For the resistance curvature κ(v) we establish the existence of infinite families for all dge 3, and we provide a quantitative lower bound for κ(v) in a simple 3-polytope in terms of the lengths of the three 2-faces incident to v. This bound leads to constructions of non-vertex-transitive, resistance-positive 3-polytopes via Δ-operations, and a degree-based obstruction showing that if each neighbor of v has degree at most d_v-2, then κ(v)le 0. Our results suggest that positive curvature on polytopal skeletons is rare and constrained.