Umbel convexity and the geometry of trees

TL;DR

This work introduces umbel $p$-convexity as a robust asymptotic metric invariant that captures the geometry of countably branching trees and provides a metric counterpart to Banach-space properties like Rolewicz's property $(\beta)$. It shows that property $(\beta_p)$ implies umbel $p$-convexity, yielding a Poincaré-type inequality that links metric geometry to renorming theory, and it extends these ideas through infrasup-umbel convexity to derive compression bounds for coarse embeddings of trees. The paper also explores stability under nonlinear quotients, furnishes a broad class of examples including infinite-dimensional Heisenberg groups, and derives Markov and diamond-type convexity results for Heisenberg groups via a parallelogram inequality. Further, it develops relaxations such as fork and infrasup-fork inequalities, connects to non-negative curvature, and discusses open problems in renorming, submetric characterizations, and the precise relationship between these invariants. Overall, the work provides a unifying, metric-analytic framework for asymptotic Banach-space geometry with concrete embedding obstructions and curvature characterizations.

Abstract

For every $p\in(0,\infty)$, a new metric invariant called umbel $p$-convexity is introduced. The asymptotic notion of umbel convexity captures the geometry of countably branching trees, much in the same way as Markov convexity, the local invariant which inspired it, captures the geometry of bounded degree trees. Umbel convexity is used to provide a ``Poincaré-type" metric characterization of the class of Banach spaces that admit an equivalent norm with Rolewicz's property $(β)$. We explain how a relaxation of umbel $p$-convexity, called infrasup-umbel $p$-convexity, plays a role in obtaining compression rate bounds for coarse embeddings of countably branching trees. Local analogues of these invariants - fork $p$-convexity and infrasup-fork $p$-convexity - are introduced, and their relationship to Markov $p$-convexity and relaxations of the $p$-fork inequality is discussed. The metric invariants are estimated for a large class of Heisenberg groups, and in particular a parallelogram $p$-convexity inequality is proved for Heisenberg groups over $p$-uniformly convex Banach spaces. Finally, a new characterization of non-negative curvature is given.

Figures (2)

• Figure 1: $\mathsf K_{1,\omega}$ in the umbel position
• Figure 2: Fork configuration

Theorems & Definitions (87)

• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4
• Definition 5
• Theorem 6
• Theorem A
• Definition 7
• Theorem B
• Theorem C