Séminaire Lotharingien de Combinatoire, 95B.9 (2026), 11 pp.
Jesú,s A. De Loera, Jillian Eddy, Sawyer Jack Robertson, and José A. Samper
Which Polytopes are Positively Curved?
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 f-numbers and establish the existence of
infinite families of Forman-Ricci-positive polytopes in every fixed
dimension d≥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 d≥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 dv-2, then κ(v) ≤ 0. Our
results suggest that positive curvature on polytopal skeletons is rare
and constrained.
Received: November 15, 2025.
Accepted: February 15, 2026.
Final version: April 1, 2026.
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