Umbel convexity and the geometry of trees

TL;DR

This paper introduces umbel convexity, a new metric invariant that characterizes the geometry of trees and Banach spaces with Rolewicz's property, with applications to embeddings, curvature, and geometric inequalities.

Contribution

It defines umbel p-convexity and infrasup-umbel p-convexity, linking them to Banach space geometry, tree embeddings, and curvature, expanding the toolkit for metric and Banach space analysis.

Findings

Introduces umbel p-convexity as a new metric invariant.

Provides a Poincaré-type characterization of Banach spaces with Rolewicz's property.

Estimates invariants for Heisenberg groups and characterizes non-negative curvature.

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 ``Poincar\'e-type" metric characterization of the class of Banach spaces that admit an equivalent norm with Rolewicz's property $(\beta)$. 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.