We investigate the geometry of sets in Euclidean and infinite-dimensional Hilbert spaces. We establish sufficient conditions that ensure a set of points is contained in the image of a (1/s)-Hölder continuous map f : [0, 1] → l 2 , with s > 1. Our results are motivated by and generalize the "sufficient half" of the Analyst's Traveling Salesman Theorem, which characterizes subsets of rectifiable curves in R N or l 2 in terms of a quadratic sum of linear approximation numbers called Jones' beta numbers. The original proof of the Analyst's Traveling Salesman Theorem depends on a well-known metric characterization of rectifiable curves from the 1920s, which is not available for higherdimensional curves such as Hölder curves. To overcome this obstacle, we reimagine Jones' non-parametric proof and show how to construct parameterizations of the intermediate approximating curves f k ([0, 1]). We then find conditions in terms of tube approximations that ensure the approximating curves converge to a Hölder curve. As an application to the geometry of measures, we identify conditions that guarantee fractional rectifiability of pointwise doubling measures in R N .

For all 1⩽m⩽n−1, we investigate the interaction of locally finite measures in Rn with the family of m‐dimensional Lipschitz graphs. For instance, we characterize Radon measures μ, which are carried by Lipschitz graphs in the sense that there exist graphs normalΓ1,normalΓ2,⋯ such that μ(double-struckRn∖⋃1∞normalΓi)=0, using only countably many evaluations of the measure. This problem in geometric measure theory was classically studied within smaller classes of measures, for example, for the restrictions of m‐dimensional Hausdorff measure Hm to E⊆double-struckRn with 0

We prove that if a simply connected nilpotent Lie group quasi-isometrically embeds into an $L^1$ space, then it is abelian. We reach this conclusion by proving that every Carnot group that bi-Lipschitz embeds into $L^1$ is abelian. Our proof follows the work of Cheeger and Kleiner, by considering the pull-back distance of a Lipschitz map into $L^1$ and representing it using a cut measure. We show that such cut measures, and the induced distances, can be blown up and the blown-up cut measure is supported on “generic” tangents of the original sets. By repeating such a blow-up procedure, one obtains a cut measure supported on half-spaces. This differentiation result then is used to prove that bi-Lipschitz embeddings can not exist in the non-abelian settings.