[Paper Review] Rectifiability of pointwise doubling measures in Hilbert Space

Lisa Naples|arXiv (Cornell University)|Feb 18, 2020

Point processes and geometric inequalities17 references4 citations

TL;DR

This paper extends the Analyst’s Traveling Salesman Theorem to Hilbert space by introducing a multiresolution family of windows and a weighted Jones’ function to characterize 1-rectifiable pointwise doubling measures. It proves that rectifiability is equivalent to uniform mass concentration around lines at all scales, and constructs rectifiable curves via δ-nets, overcoming infinite-dimensional challenges via geometric control and projection arguments.

ABSTRACT

In geometric measure theory, there is interest in studying the interaction of measures with rectifiable sets. Here, we extend a theorem of Badger and Schul in Euclidean space to characterize rectifiable pointwise doubling measures in Hilbert space. Given a measure $μ$, we construct a multiresolution family $\mathscr{C}^μ$ of windows, and then we use a weighted Jones' function $\hat{J}_2(μ, x)$ to record how well lines approximate the distribution of mass in each window. We show that when $μ$ is rectifiable, the mass is sufficiently concentrated around a lines at each scale and that the converse also holds. Additionally, we present an algorithm for the construction of a rectifiable curve using appropriately chosen $δ$-nets. Throughout, we discuss how to overcome the fact that in infinite dimensional Hilbert space there may be infinitely many $δ$-separated points, even in a bounded set. Finally, we prove a characterization for pointwise doubling measures carried by Lipschitz graphs.

Motivation & Objective

• To extend the characterization of rectifiable measures from Euclidean space to infinite-dimensional Hilbert space.
• To address the challenge that bounded sets in Hilbert space may contain infinitely many δ-separated points, complicating standard net-based constructions.
• To establish a pointwise criterion—via a weighted Jones’ function—for identifying rectifiable components of pointwise doubling measures.
• To provide an algorithm for constructing rectifiable curves through δ-nets that ensures finite length and connectedness.
• To characterize measures carried by Lipschitz graphs in Hilbert space using geometric and measure-theoretic conditions.

Proposed method

• Introduces a multiresolution family of windows $\mathscr{C}^\mu$ to localize mass distribution at different scales.
• Defines a weighted Jones’ function $\hat{J}_2(\mu,x)$ that quantifies how well lines approximate the measure’s mass in each window.
• Uses the Vitali Covering Theorem adapted to doubling measures to relate packing measure and measure via lower density assumptions.
• Applies the completeness of the Hausdorff metric space of closed sets to show convergence of nested δ-nets to a limit curve.
• Employs projection arguments and cone conditions to control the geometry of sets and ensure bi-Lipschitz control over projections.
• Constructs a rectifiable curve $\Gamma_k$ inductively from $\Gamma_{k-1}$ using bridges between δ-nets, with length estimates based on phantom length and core bridges.

Experimental results

Research questions

• RQ1 Can the characterization of rectifiable measures via the Jones function be extended from $\mathbb{R}^n$ to infinite-dimensional Hilbert space?
• RQ2 How can one construct a rectifiable curve in Hilbert space when bounded sets contain infinitely many δ-separated points?
• RQ3 What pointwise geometric conditions on a doubling measure imply that its support is rectifiable?
• RQ4 Under what conditions is a pointwise doubling measure carried by a Lipschitz graph in Hilbert space?
• RQ5 Can the Analyst’s Traveling Salesman Theorem be generalized to Hilbert space using a weighted Jones function and net-based curve construction?

Key findings

• A measure $\mu$ is 1-rectifiable if and only if the weighted Jones function $\hat{J}_2(\mu,x)$ is finite $\mu$-almost everywhere.
• The construction of $\Gamma_k$ from $\Gamma_{k-1}$ ensures connectedness and finite length, with length estimates controlled by phantom length and core bridge contributions.
• The limit curve $\Gamma = \bigcap_k \Gamma_k$ is rectifiable and carries the measure $\mu$ up to null sets.
• For a measure $\mu$ supported on a Lipschitz graph, the cone condition $|P_V x - P_V y| \geq (1 - \alpha^2)^{1/2} |x - y|$ implies bi-Lipschitz control and allows extension to the whole graph.
• The maximal cone width $\eta_\alpha$ is explicitly bounded by $\eta_\alpha = \sqrt{1 - \left(t_1 t_2 + \sqrt{(1 - t_1^2)(1 - t_2^2)}\right)}$ where $t_1 = 1 - \alpha$, $t_2 = 1 - 2\alpha$, ensuring geometric control in the construction.
• The proof overcomes infinite-dimensionality by using the completeness of the Hausdorff metric space of closed sets to ensure convergence of nested δ-nets to a limit curve.