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 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 Harmonic analysis Real analysis Geometry Theorems in discrete mathematics