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 Rademacher's theorem to metric space-valued Lipschitz functions.

Discussion

Rademacher's theorem states that a Lipschitz map ''f'' : R^''n'' → R^''m'' is differentiable almost everywhere in R^''n''; in other words, for almost every ''x'', ''f'' is approximately linear in any sufficiently small range of ''x''. If ''f'' is a function from a Euclidean space R^''n'' 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'' :  ,1nbsp;→ ''L''¹( ,1, mapping the unit interval into the space of integrable functions, defined by ''f''(''x'') = ''χ'' _{,''x''/sub>, this function is Lipschitz (and in fact, an isometry) since, if 0 ≤ ''x'' ≤ ''y''≤ 1, then}

:$, f(x)-f(y), =\int_0^1 , \chi_(t)-\chi_(t), \,dt = \int_x^y \, dt = , x-y, ,$

but one can verify that lim_''h''→0(''f''(''x'' + ''h'') −  ''f''(''x''))/''h'' does not converge to an ''L''¹ function for any ''x'' in ,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^''n'' which is a function on R^''n'' defined by the limit

: $MD(f,z)(x)=\lim_ \frac$
whenever the limit exists (here ''d'' _''X'' denotes the metric on ''X'').

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

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

The little-o notation employed here means that, at values very close to ''z'', the function ''f'' is approximately an isometry from R^''n'' with respect to the seminorm MD(''f'', ''z'') into the metric space ''X''.

References

{{Reflist

Lipschitz maps
Mathematical analysis