I will mention six different applications:
- Characterization of almost everywhere differentiability.
The following result is a consequence of the Rademacher theorem:
Theorem (Stepanov). A function $f:\Omega\to\mathbb{R}$ defined on an open set $\Omega\subset\mathbb{R}^n$ is differentiable almost everywhere if and only if $$ \limsup_{y\to x}\frac{|f(y)-f(x)|}{|y-x|}<\infty $$ for almost all $x\in\Omega$.
A beautiful proof of this classical result is given in [4].
- Uniqueness of the closest point.
Let $K\subset\mathbb{R}^n$ be a compact set. For $x\in \mathbb{R}^n\setminus K$ let $$ D_x=\{y\in K:\, d(x,y)=\operatorname{dist}(x,K)\}. $$ If the set $D_x$ consists of one point, then it means there there is a unique point in $K$ that is closes to $x$. Unfortunately there might be points where the closest point is not unique i.e. when $D_x$ contains more than one point. For example if $K$ is a sphere centered at $x$, then $D_x=K$. However, the set of non-uniqueness points is small:
Theorem. The set of points $x\in \mathbb{R}^n\setminus K$ such that the closest point in $K$ to $x$ is not unique (i.e. $D_x$ has more than one point) has measure zero.
Proof. Indeed, the distance function is Lipschitz and hence differentiable almost everywhere (by Rademacher). However, if the distance is differentiable at $x$, then $D_x$ consists of one point. For a proof of this fact see https://mathoverflow.net/a/299066/121665. $\Box$
For a related post see: Set of points with a unique closest point in a compact set.
- Aleksandrov differentiability theorem.
Aleksandrov proved the following important result about the second order differentiability of convex functions.
Theorem (Aleksandrov). Let $f:\mathbb{R}^n\to\mathbb{R}$ be a convex function. Then $f$ is locally Lipschitz and hence differentiable a.e. (Rademacher). Let $E\subset\mathbb{R}^n$ be the set of points where $f$ is differentiable. Then for almost all $x\in E$ there is a symmetric matrix $D^2f(x)$ such that $$ (1)\ \ \ \ \ \ \ \lim_{y\to x} \frac{|f(y)-f(x)-Df(x)(y-x)-\frac{1}{2}(y-x)^TD^2f(x)(y-x)|}{|y-x|^2}=0 $$ and $$ (2)\ \ \ \ \ \ \ \ \ \ \ \ \ \ \lim_{E\ni y\to x} \frac{|Df(y)-Df(x)-D^2f(x)(y-x)|}{|y-x|}=0. $$
This result can be obtained from the Rademacher theorem. The idea is as follows. The subdifferential $\partial f(x)$ is the set of all $v\in\mathbb{R}^n$ such that $$ f(y)\geq f(x)+\langle v,y-x\rangle \ \quad \text{for all $y\in\mathbb{R}^n$.} $$ It is easy to prove that $\partial f(x)\neq\emptyset$ for all $x$. Let $$ \Gamma f=\{(x,y):\, x\in\mathbb{R}^n,\ y\in \partial f(x)\} $$ be the graph of $x\mapsto\partial f(x)$. Note that this is a multi-valued function since $\partial f(x)$ may have more than one point. However, if you apply a suitable rotation of $\Gamma f$ by $45^o$, $\Gamma f$ will became a graph of a $1$-Lipschitz function $g:\mathbb{R}^n\to\mathbb{R}^n$. Clearly $g$ is differnetiable a.e. so the graph $\Gamma f$ has tangent space almost everywhere (as isometric to the graph of $g$). Since $\Gamma f$ is more or less the graph of the gradient $Df$, it follows that $Df$ is differentiable a.e. as stated in (2). This however, implies (1) too.
For more details, see
[AA] L. Ambrosio, G. Alberti, A geometrical approach to monotone functions in $\mathbb{R}^n$. Math Z. 230(1999), 259-316, DOI: 10.1007/PL00004691.
- Non-embedding of the Heisenberg group.
Theorem (Semmes). The Heisenberg group $\mathbb{H}^n$ does not admit a bi-Lipschitz embedding not any Euclidean space.
That was observed by Semmes and it follows from a version of the Rademacher functions on the Heisenberg group proved by Pansu, see [1] page 397. See also Non embedding of the Heisenberg group
- Kirchheim-Rademacher theorem.
Suppose that $f:\Omega\to\mathbb{R}^m$ (not necessarily Lipschitz) is differentiable at $x\in\Omega$. Then $$ \left|\frac{|f(y)-f(x)|-|Df(x)(y-x)|}{|y-x|}\right| \leq \frac{|f(y)-f(x)-Df(x)(y-x)|}{|y-x|} \stackrel{y\to x}{\longrightarrow} 0. $$ Observe that $\Vert z\Vert_x:=|Df(x)z|$ is a seminorm (that mens $\Vert z_1+z_2\Vert_x\leq\Vert z_1\Vert_x+\Vert z_2\Vert_x$, $\Vert tz\Vert_x=|t|\Vert z\Vert_x$, but $\Vert\cdot\Vert_x$ may vanish on a subspace of $\mathbb{R}^n$).
Theorem (Kirchheim). If $f:\mathbb{R}^n\supset\Omega\to X$ is a Lipschitz continuous mapping into any metric space $(X,d)$, then for almost all $x\in\Omega$, there is a seminorm $\Vert\cdot\Vert_x$ in $\mathbb{R}^n$ such that $$ \frac{d(f(y),f(x))-\Vert y-x\Vert_x}{|y-x|} \to 0 \quad \mbox{as $y\to x$.} $$
The seminorm $\Vert\cdot\Vert_x$ is called the metric derivative of $f$.
This result was proved in [2]. For another proof see [1].
- Topological dimension
The following result can be proved as an application of the Kirchheim-Rademacher theorem along with the Kirchheim area formula.
Theorem. Suppose that $f:\mathbb{R}^n\supset\Omega\to X$, $\Omega$ open, is a Lipschitz continuous map onto a metric space $X$, $f(\Omega)=X$. Then $\operatorname{dim} X=n$ if and only if $\mathcal{H}^n(X)>0$.
For further comments, see Topological dimension, Hausdorff dimension, and Lipschitz mappings.
- Length preserving mappings
(See also the answer of Anton Petrunin). In [5], Theorem 2.4.11, Gromov proved that any Riemannian manifold of dimension $n$ admits a mapping into $\mathbb{R}^n$ that preserves lengths of curves. Such a mapping is necessarily Lipschitz and hence differentiable almost everywhere (by Rademacher). One can prove that the Jacobian of such a mapping is different than zero almost everywhere, and hence there is no curve-length preserving mapping into $\mathbb{R}^m$ for $m < n$. More precisely we have.
Theorem. Let $M^n$ and $N^m$ be Riemannian manifolds of dimensions $n$ and $m$ respectively. If $f:M\to N$ is a mapping that preserves length of curves, then it is Lipschitz, differentiable a.e. and $\operatorname{rank} Df= n$ a.e. In particular, we necessarily have $m\geq n$.
Proof. The fact that $f$ is Lipschitz follows from the definition of the Riemannian metric. Hence it is locally Lipschitz when represented in a local coordinate system. The map represented in a coordinate system will not necessarily preserve length, but locally it will satisfy $\ell(f\circ\gamma)\geq C\ell(\gamma)$ and it easily follows that the directional derivatives will satisfy $D_vf\geq C$ a.e. so the derivative is non-degenerate a.e. so its rank is equal $n$.
References
[1] L. Ambrosio, B. Kirchheim, Bernd Rectifiable sets in metric and Banach spaces. Math. Ann. 318 (2000), 527–555.
[2] B. Kirchheim, Rectifiable metric spaces: local structure and regularity of the Hausdorff measure. Proc. Amer. Math. Soc. 121 (1994), 113-123.
[3] S. Semmes, On the nonexistence of bi-Lipschitz parameterizations and geometric problems about $A_\infty$-weights. Rev. Mat. Iberoamericana 12 (1996), 337-410.
[4] J. Maly, A simple proof of the Stepanov theorem on differentiability almost everywhere. Exposition. Math. 17 (1999), no. 1, 59–61.
[5] M. Gromov, Partial differential relations. Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], 9. Springer-Verlag, Berlin, 1986.
