相关论文: Stratified $\beta$-numbers and traveling salesman …
Let $\mathbb{G}$ be any Carnot group. We prove that, if a subset of $\mathbb{G}$ is contained in a rectifiable curve, then it satisfies Peter Jones' geometric lemma with some natural modifications. We thus prove one direction of the…
We continue to develop a program in geometric measure theory that seeks to identify how measures in a space interact with canonical families of sets in the space. In particular, extending a theorem of the first author and R. Schul in…
We show that a sufficient condition for a subset $E$ in the Heisenberg group (endowed with the Carnot-Carath\'{e}odory metric) to be contained in a rectifiable curve is that it satisfies a modified analogue of Peter Jones's geometric lemma.…
In his 1990 Inventiones paper, P. Jones characterized subsets of rectifiable curves in the plane via a multiscale sum of $\beta$-numbers. These $\beta$-numbers are geometric quantities measuring how far a given set deviates from a best…
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 $(1/s)$-H\"older continuous map $f:[0,1]\rightarrow…
In his 1990 paper, Jones proved the following: given $E \subseteq \mathbb{R}^2$, there exists a curve $\Gamma$ such that $E \subseteq \Gamma$ and \[ \mathscr{H}^1(\Gamma) \sim \text{diam}\, E + \sum_{Q} \beta_{E}(3Q)^2\ell(Q).\] Here,…
We prove a version of Peter Jones' Analyst's traveling salesman theorem in a class of highly non-Euclidean metric spaces introduced by Laakso and generalized by Cheeger-Kleiner. These spaces are constructed as inverse limits of metric…
We show that if a subset $K$ in the Heisenberg group (endowed with the Carnot-Carath\'{e}odory metric) is contained in a rectifiable curve, then it satisfies a modified analogue of Peter Jones's geometric lemma. This is a quantitative…
We are interested in quantitative rectifiability results for subsets of infinite dimensional Hilbert space $H$. We prove a version of Azzam and Schul's $d$-dimensional Analyst's Travelling Salesman Theorem in this setting by showing for any…
Jet spaces on $\mathbb R^n$ have been shown to have a canonical structure of stratified Lie groups (also known as Carnot groups). We construct jet spaces over stratified Lie groups adapted to horizontal differentiation and show that these…
We study how generalized Jones $\beta$-numbers relate to harmonic measure. Firstly, we generalize a result of Garnett, Mourgoglou and Tolsa by showing that domains in $\mathbb{R}^{d+1}$ whose boundaries are lower $d$-content regular admit…
In the setting of Carnot groups, we are concerned with the rectifiability problem for subsets that have finite sub-Riemannian perimeter. We introduce a new notion of rectifiability that is, possibly, weaker than the one introduced by…
In this Note, we define a class of stratified Lie groups of arbitrary step (that are called ``groups of type $\star$'' throughout the paper), and we prove that, in these groups, sets with constant intrinsic normal are vertical halfspaces.…
We introduce a notion of rectifiability modeled on Carnot groups. Precisely, we say that a subset E of a Carnot group M and N is a subgroup of M, we say E is N-rectifiable if it is the Lipschitz image of a positive measure subset of N.…
The ``analyst's traveling salesman theorem'' of geometric measure theory characterizes those subsets of Euclidean space that are contained in curves of finite length. This result, proven for the plane by Jones (1990) and extended to…
By considering homotopies that preserve the stratification, one obtains a natural notion of homotopy for stratified spaces. In this short note, we introduce invariants of stratified homotopy, the stratified homotopy groups. We show that…
In his 1990 Inventiones paper, P. Jones characterized subsets of rectifiable curves in the plane, using a multiscale sum of what is now known as Jones $\beta$-numbers, numbers measuring flatness in a given scale and location. This work was…
We give a constant factor approximation algorithm for the asymmetric traveling salesman problem when the support graph of the solution of the Held-Karp linear programming relaxation has bounded orientable genus.
We show that, in arbitrary Carnot groups, pliability in a subset of directions is sufficient to guarantee the existence of a Whitney-type extension and a Lusin approximation for curves with tangent vectors in the same set of directions. We…
The $k$-Opt heuristic is a simple improvement heuristic for the Traveling Salesman Problem. It starts with an arbitrary tour and then repeatedly replaces $k$ edges of the tour by $k$ other edges, as long as this yields a shorter tour. We…