Subsets of rectifiable curves in Hilbert space-the analyst’s TSP

Abstract

We study one dimensional sets (Hausdorff dimension) lying in a Hilbert space. The aim is to classify subsets of Hilbert spaces that are contained in a connected set of finite Hausdorff length. We do so by extending and improving results of Peter Jones and Kate Okikiolu for sets in R d . Their results formed the basis of quantitative rectifiability in R d . We prove a quantitative version of the following statement: a connected set of finite Hausdorff length (or a subset of one), is characterized by the fact that inside balls at most scales around most points of the set, the set lies close to a straight line segment (which depends on the ball). This is done via a quantity, similar to the one introduced in [Jon90], which is a geometric analog of the Square function. This allows us to conclude that for a given set K, the ℓ 2 norm of this quantity (which is a function of K) has size comparable to a shortest (Hausdorff length) connected set containing K. In particular, our results imply that, with a correct reformulation of the theorems, the estimates in [Jon90,Oki92] are independent of the ambient dimension. Mathematics Subject Classification (2000): 28A75

The analyst's traveling salesman theorem in graph inverse limits

“…As one final remark, note that the infinite-dimensional case of Theorem 1.1, contained in [27], shows that, if interpreted correctly, the Hilbert space analyst's traveling salesman theorem can be proven without a doubling assumption on the space. Our proofs of both Theorem 1.5 and Theorem 1.6 use the doubling property of the space, and we have not considered what happens in the non-doubling case for limits of graphs.…”

“…The following definition is taken from Lemma 3.11 of [27] (adapted to be m-adic rather than dyadic). Fix a large constant J ∈ N.…”

“…The idea of dividing the collection of balls into two sub-collections with these qualitative features is now standard (see [24], [27], [25], [22]) but our particular division is sensitive to the current context.…”

“…It was first proven in the plane by Jones [17], and then extended to R d by Okikiolu [24] and to Hilbert space by the second author [27]. In order to state it, we need to first define the Jones β numbers in the Hilbert space (or Euclidean) setting.…”

“…Let B ⊂ H be a ball in a (possibly infinite dimensional) Hilbert space H. Let E ⊂ H be any subset. Define [17,24,27]) Let H be a (possibly infinite dimensional) Hilbert space. There is a constant A 0 > 1 such that for any A > A 0 there is a constant C = C(A) > 0 such that the following holds.…”