I will mention three diffenent applications:
- 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. $\Box$
- Non-embedding of the Heisenberg group.
Theorem. 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-Rdemacher 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].
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.