A posteriori, for E to be contained in a rectifiable curve Γ, since Γ has tangents at H1-almost every point in Γ, 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: Where Q is any square, is the sidelength of Q, and dist measures the distance from x to the line L. Intuitively, is the width of the smallest rectangle containing the portion of E inside Q, and hence gives us a scale invariant notion of flatness.
Jones' traveling salesman theorem in R2
Let Δ denote the collection of dyadic squares, that is, where denotes the set of integers. For a set, define 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 can be contained in a curve with length no more than Cβ.
Conversely, if Γ is a rectifiable curve, then β < CH1.
The Traveling Salesman Theorem was shown to hold in general Euclidean spaces by Kate Okikiolu, that is, the same theorem above holds for sets, d > 1, where Δ is now the collection of dyadic cubes in defined in a similar way as dyadic squares. In her proof, the constant Cgrows exponentially with the dimension d. With some slight modifications to the definition of β, 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..
Hahlomaa further adjusted the definition of β to get a condition for when a set E of an arbitrary metric space may be contained in the Lipschitz-image of a subset of positive measure. For this, he had to redefine the definition of the β-numbers using menger curvature. 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.
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.
