Metric differential

In mathematical analysis, a metric differential is a generalization of a derivative for a Lipschitz continuous function defined on a Euclidean space and taking values in an arbitrary metric space. With this definition of a derivative, one can generalize Rademarcher's theorem to metric space-valued Lipschitz functions.

Discussion

Rademacher's theorem states that a Lipschitz map f : Rⁿ → R^m is differentiable amost everywhere in Rⁿ; in other words, for almost every x, f is approximately linear when you look in a small enough neighborhood of x. If f is a function from a Euclidean space Rⁿ that takes values instead in a metric space X, it doesn't immediately make sense to talk about differentiability since X has no linear structure a priori. Even if you assume that X is a Banach space and ask whether a Fréchet derivative exists almost everywhere, this does not hold. For example, consider the function f : [0,1] → L¹([0,1]), mapping the unit interval into the space of integrable functions, defined by f(x) = χ_[0,x], this function is Lipschitz (and in fact, an isometry) since, if 0 ≤ x ≤ y≤ 1, then

$|f(x)-f(y)|=\int_0^1 |\chi_{[0,x]}(t)-\chi_{[0,y]}(t)|\,dt = \int_x^y \, dt = |x-y|,$

but one can verify that lim_h→0(f(x + h) −  f(x))/h does not converge to an L¹ function for any x in [0,1], so it is not differentiable anywhere.

However, if you look at Rademacher's theorem as a statement about how a Lipschitz function stabilizes as you zoom in on almost every point, then such a theorem exists but is stated in terms of the metric properties of f instead of its linear properties.

Definition and existence of the metric differential

A substitute for a derivative of f:Rⁿ → X is the metric differential of f at a point z in R^mn which is a function on R^m defined by the limit

$MD(f,z)(x)=\lim_{r\rightarrow 0} \frac{d_{X}(f(z+rx),f(z))}{r}$

whenever the limit exists (here d _X denotes the metric on X).

A theorem due to Bernd Kirchheim^[1] states that a Rademacher theorem in terms of metric differentials holds: for almost every z in R^m, MD(f, z) is a seminorm and

$d_X(f(x),f(y)) - MD(f,z)(x-y) = o(|x-z|+|y-z|). \,$

What this little-o notation means is that, at values very close to z, the function f is approximately an isometry from Rⁿ with respect to the seminorm MD(f, z) into the metric space X.

References

1. ^ Kirchheim, Bernd (1994). "Rectifiable metric spaces: local structure and regularity of the Hausdorff measure". Proc. of the Am. Math. Soc. 121: 113–124.