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 formula_1 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 rectifiable curve has tangents at almost all of  its points, where in this case "almost all" means all but a subset whose one-dimensional Hausdorff measure is zero. Accordingly, if a set is contained in a rectifiable curve, the set must look "flat" when zooming in on almost all of its points. This suggests that testing us whether a set could be contained in a rectifiable curve must somehow incorporate information about how flat it is when one zooms in on its points at different scales. This discussion motivates the definition of the following quantity: Where formula_2 is the set that is to be contained in a rectifiable  curve, formula_3 is any square, formula_4 is the side length of formula_3, and distformula_6 measures the distance from formula_7 to the line formula_8. Intuitively, formula_9 is the width of the smallest rectangle containing the portion of formula_2 inside formula_3, and hence formula_12 gives a scale invariant notion of "flatness". Jones' traveling salesman theorem in R. Let Δ denote the collection of dyadic squares, that is, where formula_14 denotes the set of integers. For a set formula_15, define where diam "E" is the diameter of "E". Then Peter Jones's analyst's traveling salesman theorem may be stated as follows: 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 formula_17, "d" > 1, where Δ is now the collection of dyadic cubes in formula_18 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 formula_19 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.