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'rectifiable set'
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| Title of object: |
rectifiable set |
| Canonical Name: |
RectifiableSet |
| Type: |
Definition |
| Created on: |
2004-07-09 11:26:20 |
| Modified on: |
2004-07-14 06:50:18 |
| Classification: |
msc:49Q15 |
| Defines: |
density, tangent vector, approximate tangent plane |
Preamble:
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Content:
A set $S\subset \R^n$ is said to be \emph{$(\H^m,m)$-rectifiable} or simply \emph{$m$-rectifiable} if $\H^m(S)<+\infty$ and there exists a countable sequence of Lipschitz continuous functions $f_k\colon \R^m \to \R^n$ such that
\H^m(S\setminus \bigcup_k f_k(\R^m)=0.
Given a point $x\in S$ one can define the \emph{density} of $S$ in $x$ as
\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 $\R^m$. Notice that an $m$-dimensional plane has density $1$ in all points.
It turns out that if $S$ is rectifiable, then in $\H^m$-a.e.\ point $x\in S$ the density $\Theta^m(S,x)$ exists and is equal to $1$. Moreover in $\H^m$-a.e.\ point $x\in S$ there exists an \emph{approximate tangent plane} to $S$ as defined below.
Given a point $x\in S\subset \R^n$ and a vector $v\in \R^n$ we say that $v$ is tangent to $S$ in $x$ if there exists a sequence of points $x_k\in S$, $x_k\to x$ and a sequence of real numbers $\lambda_k$ such that
\lim_{k\to \infty} \lambda_k (x_k -x) = v.
We say that a vector $v$ is \emph{approximately tangent} to $S$ in $x$ if it is
a tangent vector to every subset $S'$ of $S$ such that $\Theta^m(S\setminus S',x)=0$.
The \emph{approximate tangent cone} to $S$ in $x$ is the set of all approximately tangent vectors to $S$ in $x$. If the approximate tangent cone is an $m$-dimensional vector subspace of $\R^n$, it is called the \emph{approximate tangent plane}.
Notice that if $S\subset \R^n$ is any $m$-dimensional regular surface,
and $Q$ is the set of all points of $\R^n$ with rational coordinates, then the set $S\cup Q$ is an $m$-rectifiable set since $\H^1(Q)=0$. Notice, however, that $\overline{S\cup Q} \subset \overline Q = \R^n$ and consequently every vector $v$ 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 $S$, because the set $Q$ has density $0$ everywhere. |
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