Rectifiable set

Summary

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.

DefinitionEdit

A Borel subset   of Euclidean space   is said to be  -rectifiable set if   is of Hausdorff dimension  , and there exist a countable collection   of continuously differentiable maps

 

such that the  -Hausdorff measure   of

 

is zero. The backslash here denotes the set difference. Equivalently, the   may be taken to be Lipschitz continuous without altering the definition.[1][2][3] Other authors have different definitions, for example, not requiring   to be  -dimensional, but instead requiring that   is a countable union of sets which are the image of a Lipschitz map from some bounded subset of  .[4]

A set   is said to be purely  -unrectifiable if for every (continuous, differentiable)  , one has

 

A standard example of a purely-1-unrectifiable set in two dimensions is the cross-product of the Smith–Volterra–Cantor set times itself.

Rectifiable sets in metric spacesEdit

Federer (1969, pp. 251–252) gives the following terminology for m-rectifiable sets E in a general metric space X.

  1. E is   rectifiable when there exists a Lipschitz map   for some bounded subset   of   onto  .
  2. E is countably   rectifiable when E equals the union of a countable family of   rectifiable sets.
  3. E is countably   rectifiable when   is a measure on X and there is a countably   rectifiable set F such that  .
  4. E is   rectifiable when E is countably   rectifiable and  
  5. E is purely   unrectifiable when   is a measure on X and E includes no   rectifiable set F with  .

Definition 3 with   and   comes closest to the above definition for subsets of Euclidean spaces.

NotesEdit

  1. ^ Simon 1984, p. 58, calls this definition "countably m-rectifiable".
  2. ^ "Rectifiable set", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
  3. ^ Weisstein, Eric W. "Rectifiable Set". MathWorld. Retrieved 2020-04-17.
  4. ^ Federer (1969, pp. 3.2.14)

ReferencesEdit

External linksEdit

  • Rectifiable set at Encyclopedia of Mathematics