**I will mention seven different applications:**
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- **Characterization of almost everywhere differentiability.**
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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].
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- **Uniqueness of the closest point.**
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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: https://mathoverflow.net/q/342308/121665.
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- **Aleksandrov differentiability theorem.**
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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](https://doi.org/10.1007/PL00004691).
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- **Non-embedding of the Heisenberg group.**
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> **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 https://mathoverflow.net/q/297806/121665
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- **Kirchheim-Rademacher theorem.**
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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].
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- **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 https://mathoverflow.net/q/391839/121665.
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- **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 $n$.
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References
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**[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.