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 which plane sets are subsets of rectifiable curves of finite length. Whereas the original traveling salesman problem asks for the shortest way to visit every vertex in a finite set with a discrete path, this analytical version may require the curve to visit 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, for a plane set : where is the set that is to be contained in a rectifiable curve, is any square, is the side length of , and dist measures the distance from to the line . Intuitively, is the width of the smallest rectangle containing the portion of inside , and hence gives 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 and is the square with same center as with side length . Then Peter Jones's 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 , 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 C grows exponentially with the