Rectifiable sets and approximate tangent planes#

Rectifiable sets are the measure theoretic counterpart to manifolds.

Definition 22 (Rectifiable set)

A \(\mathcal H^n\)-measurable set \(E\subset \mathbb R^m\) is \(n\)-rectifiable if there exist Lipschitz \(f_i \colon \mathbb R^n \to \mathbb R^m\) such that

\[\mathcal H^n\left(E\setminus \bigcup_{i\in\mathbb N} f_i(\mathbb R^n)\right)=0.\]

We will show that \(n\)-rectifiable sets possess a unique approximate \(n\)-dimensional tangent plane at almost every point.

Given \(V\in G(m,n)\), \(a\in \mathbb R^n\) and \(0<s<1\) define the cone around \(V\) centred at \(a\) with aperture \(s\) as

\[C(a,V,s)=\{x\in \mathbb R^n : \operatorname{dist}(x-a, V)< s\|x-a\|\}.\]

Definition 23 (Approximate tangent plane)

Let \(A\subset \mathbb R^m\) and \(a\in A\). A \(V\in G(m,n)\) is an approximate tangent plane to \(A\) at \(a\) if

(22)#\[\limsup_{r\to 0} \frac{\mathcal H^n(A \cap B(a,r))}{r^n}>0\]

and, for every \(0<s<1\),

(23)#\[\lim_{r\to 0} \frac{\mathcal H^n(A \cap B(a,r) \setminus C(a,V,s))}{r^n}=0.\]

Rademacher’s theorem gives a candidate for the approximate tangent plane to a rectifiable set. There are three steps required to prove that the derivative is indeed an approximate tangent plane: show that the derivative has full rank at almost every point; prove the density condition (22); and show that the sets from other parametrisations of the rectifiable set do not destroy the approximation by a tangent plane at almost every point.

The second and third steps follow from the results of the previous section. For the first step we use the following.

Lemma 7 (Easy Sard’s theorem)

If \(f\colon \mathbb R^n \to \mathbb R^m\) is Lipschitz then

\[\mathcal H^n(\{f(x): \operatorname{rank} Df(x)<n\})=0.\]

Proof. Let \(L=\operatorname{Lip}f\). Fix \(0<R<\infty\), \(\delta,\epsilon>0\) and let

\[A=\{x\in B(x,R): \operatorname{rank} Df(x)<n\}.\]

For \(x\in A\) let

\[W_x = f(x)+Df(x)(\mathbb R^n).\]

Then for sufficiently small \(0<r_x<\delta\),

\[f(B(x,r_x))\subset B(f(x),Lr_x) \cap \{y: \operatorname{dist}(y,W_x)< \epsilon r_x\}.\]

Since \(\operatorname{rank}Df(x)<n\), the set on the right hand side can be covered by \((L/\epsilon)^{n-1}\) cubes of side length \(\epsilon r_x\).

Since \(A\) is covered by balls of the form \(B(x,r_x/5)\), there exists a disjoint collection of balls \(B(x_i,r_i/5)\) such that \(A\) is covered by the union of the \(B(x_i,r_i)\). Then

\[f(A) \subset f\left(\bigcup_{i\in\mathbb N}B(x_i,r_i)\right) \subset \bigcup_{i\in\mathbb N} f(B(x_i,r_i)).\]

By the previous argument, each factor of the right hand side is covered by \((L/\epsilon)^{n-1}\) cubes of side length \(\epsilon r_i\). Thus

\[\mathcal H^n_{2\delta}(f(A)) \leq \sum_{i\in\mathbb N} \left(\frac{L}{\epsilon}\right)^{n-1} (\epsilon r_i)^{n-1} = L^{n-1}\epsilon \sum_{i\in\mathbb N}r_i^n.\]

However, the \(B(x_i,r_i/5)\) are disjoint subsets of \(B(0,R+\delta)\subset \mathbb R^n\) and so

\[\sum_{i\in\mathbb N}\left(\frac{r_i}{5}\right)^n \leq (R+\delta)^n.\]

Since \(\epsilon>0\) is arbitrary, this implies that \(\mathcal H^n_{2\delta}(f(A))=0\) and hence \(\mathcal H^n(f(A))=0\). Taking a countable union over \(R\to\infty\) completes the proof.

Lemma 8

Let \(f\colon \mathbb R^n\to \mathbb R^m\) be Lipschitz and \(S\subset \mathbb R^n\). Suppose that there exists a \(\delta>0\) such that, for each \(x,y\in S\),

\[\|f(x)-f(y)\|\geq \delta \|x-y\|.\]

Then \(f(S)\) has a unique approximate tangent plane at \(\mathcal H^n\) almost every point.*

Proof. By Lemma 7, we may suppose \(\operatorname{rank}Df(x)=n\) for every \(x\in S\). By Lemma 5, we may suppose \(\Theta^{*,n}(E,f(x))>0\) for every \(x\in S\). Fix \(x\in S\) and \(0<s<1\). There exists \(\epsilon>0\) such that

\[\|f(y)-f(x) -Df(x)(y-x)\| < s\|y-x\|\]

for all \(y\in B(x,\epsilon)\cap S\). Moreover, if \(y\in S\setminus B(x,\epsilon)\) then

\[\|f(y)-f(x)\| \geq \delta \epsilon.\]

That is, if \(a=f(x)\) and \(b=f(y)\) with \(\|a-b\|\leq \delta\epsilon\) and \(V=a+Df(x)(\mathbb R^n)\),

\[\operatorname{dist}(b-a, V) < s\|y-x\| \leq s\|b-a\|/\delta.\]

Therefore, \(V\) is an approximate tangent plane to \(f(S)\) at \(a\).

This approximate tangent plane is unique at any density point \(x\) of \(S\). Indeed, if \(V'\neq V\), let \(v\in V\setminus V'\) and let \(0<s<1\) be such that \(C(f(x),\mathbb Rv,s)\cap C(f(x),V',s)= \{0\}\). Since \(\operatorname{rank}Df(x)=n\) and \(x\) is a density point of \(S\), for sufficiently small \(r>0\) there exists \(y\in S \cap B(x,r)\) such that \(B(f(y),s r)\cap C(f(x),V',s) = \emptyset\) and

\[\frac{\mathcal H^n(B(f(y),s r)\cap S)}{r^n} \geq \delta s.\]

In particular, \(V'\) is not an approximate tangent to \(f(S')\) at \(x\).

Theorem 14

Let \(E\subset \mathbb R^m\) be \(n\)-rectifiable with \(\mathcal H^n(E)<\infty\). Then for \(\mathcal H^n\)-a.e. \(x\in E\), \(E\) has a unique approximate tangent plane at \(x\).

Proof. Let \(f\colon \mathbb R^n\to \mathbb R^m\) be one of the Lipschitz functions as in the definition of a rectifiable set and let \(S=f^{-1}(E)\). It suffices to prove that \(E\) has a unique approximate tangent plane at \(f(x)\) for \(\mathcal L^n\)-a.e. \(x\in S\).

By Lemma 7, we may suppose that \(\dim Df(\mathbb R^n)=n\) for all \(x\in S\). Fix such an \(x\) and let \(0<\epsilon<\|Df(x)^{-1}\|/2\). There exists \(\delta>0\) such that

\[\|f(y)-f(x) -Df(x)(y-x)\| < \epsilon\|y-x\|\]

for all \(y\in B(x,\delta)\cap S\). In particular, by the triangle inequality,

\[\|f(y)-f(x) \| > \epsilon\|y-x\|/2.\]

Therefore, the sets

\[S_\epsilon:=\{x\in G: \|f(y)-f(x) \| > \epsilon \|y-x\|\ \forall y\in B(x,\epsilon)\}\]

are Borel and monotonically increase to \(S\) as \(\epsilon\to 0\). Therefore it suffices to prove the result for \(\mathcal L^n\)-a.e. \(x\) in some fixed \(S_\epsilon\). Cover \(S_\epsilon\) by finitely many balls \(B_1,B_2,\ldots,B_N\) of radius \(\epsilon\). It suffices to prove the result for \(\mathcal L^n\)-a.e. \(x\) in some fixed \(S':=S_\eta\cap B_i\).

However, \(S'\) satisfies the hypotheses of Lemma 8 and so \(f(S')\) has a unique approximate tangent at \(\mathcal H^n\) almost every point. To see that this tangent is a unique approximate tangent to \(E\) at \(\mathcal H^n\) almost every point, we simply use Lemma 6: for \(\mathcal H^n\)-a.e. \(x\in f(S')\), \(\Theta^{*,n}(E\setminus f(S'),x)=0\).

In Theorem 16 we will see that the converse to Theorem 14 holds.

For \(V\in G(n,m)\), write \(\pi_V\) for the orthogonal projection onto \(V\) and equip \(G(n,m)\) with the metric \(d(V,W)=\|\pi_V-\pi_W\|\). We will consider \(\mathcal L^n\) on an element of \(G(n,m)\).

Lemma 9

Let \(f\colon \mathbb R^m \subset \mathbb R^n \to \mathbb R^m\) be Lipschitz and let \(S\subset \mathbb R^n\) satisfy \(\mathcal L^n(S)>0\). For \(\epsilon>0\) suppose that there exists an invertible linear \(L\colon \mathbb R^n\to \mathbb R^m\) such that, for all \(x,y \in S\),

\[\|f(x)-f(y)-L(x-y)\| < \frac{\epsilon}{\|L^{-1}\|}\|x-y\|.\]

Then for any \(V\in G(n,m)\) with \(\|(\pi_V|_{L(\mathbb R^n)})^{-1}\|^{-1} \geq 2\epsilon\), \(\mathcal L^n(\pi_V(f(S)))>0\).*

Proof. For any \(V\in G(n,m)\),

\[\|\pi_V(f(x)-f(y)) - \pi_V( L(x-y))\| < \epsilon\|L^{-1}\|^{-1}\|x-y\|\]

and so, if \(\|(\pi_V|_{L(\mathbb R^n)})^{-1}\|^{-1} \geq 2\epsilon\),

\[\begin{split}\|\pi_V(f(x)-f(y))\| &\geq \|\pi_V( L(x-y))\| - \epsilon\|L^{-1}\|^{-1}\|x-y\|\\ &\geq \|(\pi_V|_{L(\mathbb R^n)})^{-1}\|^{-1}\|L(x-y)\|-\epsilon\|L(x-y)\|\\ &\geq \epsilon \|L(x-y)\|\\ &\geq \epsilon \|L^{-1}\|\|x-y\|.\end{split}\]

Thus \(\pi_V\circ f\) has Lipschitz inverse on \(S\) and hence \(\mathcal L^n(\pi_V(f(S)))>0\) by Example 5.

Corollary 2

Let \(E\subset \mathbb R^m\) be \(n\)-rectifiable with \(\mathcal H^n(E)>0\). Then there exists \(W\in G(m-n,m)\) such that \(\pi_V(E)>0\) for all \(V\in G(n,m)\) with \(V\cap W=\{0\}\).*

Remark 5

The set of \(V\) that satisfy the conclusion of Corollary 2 is very large; try some examples in reasonable dimensions.

Proof. Since \(E\subset \mathbb R^m\) is rectifiable with \(\mathcal H^n(E)>0\), there exists a Lipschitz \(f\colon \mathbb R^n\to \mathbb R^m\) with \(\mathcal H^n(E\cap f(\mathbb R^n))>0\). In particular, \(S:= f^{-1}(E)\) satisfies \(\mathcal L^n(S)>0\). By Lemma 7, for \(\mathcal L^n\)-a.e. \(x\in S\), \(Df(x)\) is injective.

Fix \(\epsilon>0\) For \(M>0\), the set of invertible \(L\colon \mathbb R^n\to \mathbb R^m\) with \(\|L^{-1}\|<M\) may be covered by countably many sets of diameter \(\epsilon/M\). Varying \(M\in\mathbb N\), we see that \(\mathcal L^n\) almost all of \(S\) is covered by countably many sets of the form

\[\{x\in S: \|Df(x)-L\|< \epsilon/2\|L^{-1}\|\}.\]

Moreover, each of these sets may be covered by countably many sets of the form

\[\{x\in S: \|f(x)-f(y)-L(x-y)\| < \epsilon\|x-y\|/\|L^{-1}\|\ \forall y\in B(x,\epsilon)\}.\]

Finally, these sets may be covered by countably many sets of diameter \(\epsilon\). Therefore, for each \(j\in\mathbb N\) there exists \(S_j\subset S\) and invertible \(L_j\colon \mathbb R^n\to\mathbb R^m\) such that, for all \(x,y\in S_j^\epsilon\),

\[\|f(x)-f(y)-L_j(x-y)\| < \epsilon\|x-y\|/\|L_j\|^{-1},\]

and \(\mathcal L^n(S\setminus \bigcup_{j\in \mathbb N}S_j)=0\).

Since \(\mathcal L^n(S)>0\), there exists \(j\in \mathbb N\) with \(\mathcal L^n(S_j)>0\). Then \(S_j\) satisfies the hypotheses of Lemma 9 and so \(\mathcal L^n(\pi_V(S))\geq \mathcal L^n(\pi_V(S_j))>0\) for all \(V\in G(n,m)\) with \(\|(\pi_V|_{L_j(\mathbb R^n)})^{-1}\|^{-1} \geq 2\epsilon\). Let \(L_\epsilon=L_j\). Repeat this for each \(i\in\mathbb N\) with \(\epsilon =1/i\). The set \(G(n,m)\) is compact and so we may suppose that \(L_{1/i}(\mathbb R^n) \to W\in G(n,m)\). The only \(V\in G(n,m)\) for which \(\mathcal L^n(\pi_V(S))=0\) satisfy \(\|(\pi_V|_{L_{1/i}(\mathbb R^n)})^{-1}\|^{-1} < 2/i\) and hence \(\|(\pi_V|_{W})^{-1}\|^{-1} < 2/i\) for each \(i\in \mathbb N\). That is, \(V\cap W^{\perp} \neq \{0\}\) as required.

Exercises#

Example 46

Let \(X\) be a metric space, \(Y\subset X\) and \(f\colon Y \to \mathbb R\) \(L\)-Lipschitz. Define \(\tilde f\colon X \to \mathbb R\) by

\[\tilde f(x)= \sup\{f(y)-Ld(x,y):y\in Y\}.\]
  1. Show that \(\tilde f\) is an \(L\)-Lipschitz extension of \(f\) to \(X\). This is called the McShane–Whitney extension theorem

  2. If \(f\colon Y\to \mathbb R^n\) is \(L\)-Lipschitz, show that there is a \(\sqrt{n}L\)-Lipschitz extension of \(f\) to \(X\).

  3. The following example shows that the vector valued extension cannot have the same Lipschitz constant in general: Let

    \[Y=\{(-1,1),(1,-1),(1,1)\} \subset \ell_\infty^2\]

    and define

    \[f(-1,1)=(-1,0), \quad f(1,-1)=(1,0), \quad f(1,1)=(0,\sqrt{3}).\]

    Show that \(f\) is 1-Lipschitz but has no 1-Lipschitz extension to \(Y\cup \{(0,0)\}\).

  4. However, the Kirszbraun extension theorem states that any Lipschitz map between any two Hilbert spaces may be extended whilst preserving the Lipschitz constant.

Example 47

  1. Let \(E\subset \mathbb R^m\) be \(n\)-rectifiable. Show that \(\mathcal H^n|_E\) is \(\sigma\)-finite.

  2. Show that Theorem 14 may not be true if \(E\) does not satisfy \(\mathcal H^n(E)<\infty\).