Bidimensionality

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,

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

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

$\beta (E)=\text{diam}E+\sum _{Q\in \mathrm{\Delta }}{\beta }_E(3Q{)}^2\ell (Q)$

where diam E is the diameter of E. Then Peter Jones' 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, 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 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 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.

References

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