The traveling salesman problem in the Heisenberg group: upper bounding curvature

Abstract

We show that if a subset $K$ in the Heisenberg group (endowed with the Carnot-Carathéodory metric) is contained in a rectifiable curve, then it satisfies a modified analogue of Peter Jones's geometric lemma. This is a quantitative version of the statement that a finite length curve has a tangent at almost every point. This condition complements that of \cite{FFP} except a power 2 is changed to a power 4. Two key tools that we use in the proof are a geometric martingale argument like that of \cite{Schul-TSP} as well as a new curvature inequality in the Heisenberg group.

36 pages. Final version to appear in Transactions of the American Mathematical Society

References in corpus (2)

• An upper bound for the length of a Traveling Salesman path in the Heisenberg group
• Markov convexity and nonembeddability of the Heisenberg group

Cited by in corpus (1)

• An upper bound for the length of a Traveling Salesman path in the Heisenberg group