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rectifiable set

(Definition)

Let us denote with $\H ^m$ the -dimensional Hausdorff measure in $\mathbb{R}^n$ .

A set $S\subset \mathbb{R}^n$ is said to be countably -rectifiable if there exists a countable sequence of Lipschitz continuous functions $f_k\colon\mathbb{R}^m \to \mathbb{R}^n$ such that

$\displaystyle S\subset \bigcup_k f_k(\mathbb{R}^m).$

A set $S\subset \mathbb{R}^n$ is said to be countably $(\H ^m,m)$ -rectifiable if there exists a set which is countable -rectifiable and such that $\H ^m(S\setminus S')=0$ .

A set $S\subset \mathbb{R}^n$ is said to be $(\H ^m,m)$ -rectifiable or simply -rectifiable if it is $(\H ^m,m)$ -rectifiable and $\H ^m(S)<+\infty$ .

If is any Borel subset of $\mathbb{R}^n$ and $x\in \mathbb{R}^n$ is given, one can define the density of in as

$\displaystyle \Theta^m(S,x) = \lim_{\rho \to 0} \frac{\H ^m(S\cap B_\rho(x))} {\omega_m \rho^m}$

where $\omega_m$ is the Lebesgue measure of the unit ball in $\mathbb{R}^m$ . Notice that an

-dimensional plane $\Pi$ has density

in all points $x\in\Pi$ and density 0 in all points $x\not\in\Pi$ .

It turns out that if is rectifiable, then in $\H ^m$ -a.e. point $x\in S$ the density $\Theta^m(S,x)$ exists and is equal to . Moreover in $\H ^m$ -a.e. point $x\in S$ there exists an approximate tangent plane to as defined below.

Given a point $x\in S\subset \mathbb{R}^n$ and a vector $v\in \mathbb{R}^n$ we say that is tangent to in if there exists a sequence of points $x_k\in S$ , $x_k\to x$ and a sequence of positive numbers $\lambda_k$ such that

$\displaystyle \lim_{k\to \infty} \lambda_k (x_k -x) = v.$

is an

-dimensional manifold, then the set of tangent vectors to a point $x\in S$ is nothing else than the usual tangent plane to

We say that a vector is approximately tangent to in if it is a tangent vector to every subset of such that $\Theta^m(S\setminus S',x)=0$ . Notice that every tangent vector is also an approximately tangent vector while the converse is not always true, as it is shown in an example below. The point, here, is that being the set defined $\H ^m$ -almost everywhere, we need a stronger definition for tangent vectors.

The approximate tangent cone to in is the set of all approximately tangent vectors to in (notice that if is a tangent vector then $\lambda v$ is also a tangent vector, for all $\lambda>0$ ). If the approximate tangent cone is an -dimensional vector subspace of $\mathbb{R}^n$ , it is called the approximate tangent plane.

Notice that if $S\subset \mathbb{R}^n$ is any -dimensional regular surface, and is the set of all points of $\mathbb{R}^n$ with rational coordinates, then the set $S\cup Q$ is an -rectifiable set since $\H ^1(Q)=0$ . Notice, however, that $\overline{S\cup Q} = \mathbb{R}^n$ and consequently every vector is tangent to $S\cup Q$ in every point $x\in S\cup Q$ . On the other hand the approximately tangent vectors to $S\cup Q$ are only the tangent vectors to , because the set has density 0 everywhere.

Bibliography

1: Frank Morgan: Geometric Measure Theory: A Beginner's Guide.

"rectifiable set" is owned by paolini.

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Also defines:

density, tangent vector, approximate tangent plane

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Cross-references: coordinates, rational, surface, regular, vector subspace, cone, converse, subset, tangent plane, manifold, numbers, positive, tangent, vector, rectifiable, points, plane, unit ball, Lebesgue measure, Borel subset, functions, Lipschitz continuous, sequence, countable, Hausdorff measure
There are 21 references to this entry.

This is version 8 of rectifiable set, born on 2004-07-09, modified 2005-03-31.
Object id is 5991, canonical name is RectifiableSet.
Accessed 6646 times total.

Classification:

AMS MSC:

49Q15 (Calculus of variations and optimal control; optimization :: Manifolds :: Geometric measure and integration theory, integral and normal currents)

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