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

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 (also published in this issue), this completes the proof of the necessary half of the analyst’s traveling salesman theorem with sharp exponent in uniformly convex spaces.