Analyst's traveling salesman theorem

The analyst's traveling salesman problem is an analog of the traveling salesman problem in combinatorial optimization. In its simplest and original form, it asks under what conditions may a set E in two-dimensional Euclidean space $\mathbb{R}^2$ be contained inside a rectifiable curve of finite length. So while in the original traveling salesman problem, one asks for the shortest way to visit every vertex in a graph with a discrete path, this analytical version requires the curve to visit perhaps infinitely many points.

β-numbers

A posteriori, for E to be contained in a rectifiable curve Γ, since Γ has tangents at H¹-almost every point in Γ (where H¹ denotes one-dimensional Hausdorff measure), E must look flat when you zoom in on points in E. This suggests that a condition that would tell us whether a set could be contained in a curve must somehow incorporate information about how flat E is when we zoom in on points of E at different scales.

This discussion motivates the definition of the following quantity:

$\beta_{E}(Q)=\frac{1}{\ell(Q)}\inf\{\delta:\text{ there is a line }L\text{ so that for every }x\in E\cap Q, \; \text{dist}(x,L)<\delta\},$

Where Q is any square, $\ell(Q)$ is the sidelength of Q, and dist(x, L) measures the distance from x to the line L. Intuitively, $2\beta_E(Q)\ell(Q)$ is the width of the smallest rectangle containing the portion of E inside Q, and hence $\beta_E(Q)$ gives us a scale invariant notion of flatness.

Jones' traveling salesman theorem in R²

Let Δ denote the collection of dyadic squares, that is,

$\Delta=\{[i2^{k},(i+1)2^{k}]\times[j2^{k},(j+1)2^{k}]: i,j,k\in\mathbb{Z}\},$

where $\mathbb{Z}$ denotes the set of integers. For a set $E\subseteq\mathbb{R}^2$, define

$\beta(E)=\text{diam} E+ \sum_{Q\in\Delta}\beta_{E}(3Q)^2 \ell(Q)$

where diam E is the diameter of E. Then Peter Jones'^[1] analyst's traveling salesman theorem may be stated as follows:

• There is a number C > 0 such that whenever E is a set with such that β(E) < ∞, E can be contained in a curve with length no more than Cβ(E).

• Conversely (and substantially more difficult to prove), if Γ is a rectifiable curve, then β(Γ) < CH¹(Γ).

Generalizations and Menger curvature

Euclidean space and Hilbert space

The Traveling Salesman Theorem was shown to hold in general Euclidean spaces by Kate Okikiolu,^[2] that is, the same theorem above holds for sets $E\subseteq\mathbb{R}^d$, d > 1, where Δ is now the collection of dyadic cubes in $\mathbb{R}^d$ defined in a similar way as dyadic squares. In her proof, the constant C grows exponentially with the dimension d.

With some slight modifications to the definition of β(E), Raanan Schul^[3] showed Traveling Salesman Theorem also holds for sets E that lie in any Hilbert Space, and in particular, implies the theorems of Jones and Okikiolu, where now the constant C is independent of dimension. (In particular, this involves using β-numbers of balls instead of cubes).

Menger curvature and metric spaces

Hahlomaa^[4] further adjusted the definition of β(E) to get a condition for when a set E of an arbitrary metric space may be contained in the Lipschitz-image of a subset $A\subseteq\mathbb{R}$ of positive measure. For this, he had to redefine the definition of the β-numbers using menger curvature (since in a metric space there isn't necessarily a notion of a cube or a straight line).

Menger curvature, as in the previous example, can be used to give numerical estimates that determine whether a set contains a rectifiable subset, and the proofs of these results frequently depend on β-numbers.

Denjoy–Riesz theorem

The Denjoy–Riesz theorem gives general conditions under which a point set can be covered by the homeomorphic image of a curve. This is true, in particular, for every compact totally disconnected subset of the Euclidean plane. However, it may be necessary for such an arc to have infinite length, failing to meet the conditions of the analyst's traveling salesman theorem.

References

1. Jones, Peter (1990). "Rectifiable sets and the Traveling Salesman Problem". Inventiones Mathematicae 102: 1–15. doi:10.1007/BF01233418.
2. Okikiolu, Kate (1992). "Characterization of subsets of rectifiable curves in Rn". Journal of the London Mathematical Society 46: 336–348. doi:10.1112/jlms/s2-46.2.336.
3. Schul, Raanan (2007). "Subsets of Rectifiable curves in Hilbert Space—The Analyst's TSP". Journal d'Analyse Mathématique 103: 331–375. doi:10.1007/s11854-008-0011-y.
4. Hahlomaa, Immo (2005). "Menger curvature and Lipschitz parametrizations in metric spaces". Fund. Math. 185: 143–169. doi:10.4064/fm185-2-3.