In mathematics, a rectifiable set is a set that is smooth in a certain measure-theoretic sense.
It is an extension of the idea of a rectifiable curve to higher dimensions; loosely speaking, a rectifiable set is a rigorous formulation of a piece-wise smooth set.
As such, it has many of the desirable properties of smooth manifolds, including tangent spaces that are defined almost everywhere.
Rectifiable sets are the underlying object of study in geometric measure theory.
of Euclidean space
, and there exist a countable collection
of continuously differentiable maps such that the
The backslash here denotes the set difference.
may be taken to be Lipschitz continuous without altering the definition.
[1][2][3] Other authors have different definitions, for example, not requiring
is a countable union of sets which are the image of a Lipschitz map from some bounded subset of
, one has A standard example of a purely-1-unrectifiable set in two dimensions is the Cartesian product of the Smith–Volterra–Cantor set times itself.
251–252) gives the following terminology for m-rectifiable sets E in a general metric space X.
comes closest to the above definition for subsets of Euclidean spaces.