Subsets of rectifiable curves in Banach spaces II: universal estimates for almost flat arcs

TL;DR

This paper establishes that in Banach spaces, the regions where rectifiable curves appear nearly flat are quantitatively small, contributing to a comprehensive proof of the Analyst's Traveling Salesman theorem with sharp exponents in uniformly convex spaces.

Contribution

It provides universal estimates for almost flat arcs in Banach spaces, completing the proof of the Analyst's Traveling Salesman theorem with sharp exponents in uniformly convex spaces.

Findings

Quantitative bounds on almost flat arcs in Banach spaces

Universal constants independent of curve and space dimension

Completes the proof of the Analyst's Traveling Salesman theorem in certain spaces

Abstract

We prove that in any Banach space the set of windows in which a rectifiable curve resembles two or more straight line segments is quantitatively small with constants that are independent of the curve, the dimension of the space, and the choice of norm. Together with Part I, this completes the proof of the necessary half of the Analyst's Traveling Salesman theorem with sharp exponent in uniformly convex spaces.