Metric differential

The metric differential is a replacement for the derivation term for mappings in metric spaces . It was introduced in 1994 by the German mathematician Bernd Kirchheim in an essay on the regularity of Hausdorff measures . The main application of the metric differential is the generalization of Rademacher's theorem of functions between Euclidean spaces to those in general metric spaces.

motivation

In addition to their metric structure, Euclidean spaces also have a linear one . Therefore it is possible to consider local linear approximations for a function between Euclidean spaces . If there is a best such approximation for a point in the domain of definition , the function there is called (totally) differentiable and the corresponding linear function is called derivative or differential at this point. The derivation in a certain direction can also be viewed restrictively . For mappings in general metric spaces, such statements cannot initially be made, since the said linear structure is missing. The metric differential now serves to transfer these terms to the last-mentioned figures in the sense of a best isometric approximation.

definition

Furthermore, let be a function from a Euclidean space into a metric space and a point. Sit now ${\displaystyle f\colon \mathbb {R} ^{n}\to (X,d_{X})}$ ${\displaystyle X}$ ${\displaystyle x\in \mathbb {R} ^{n}}$

${\displaystyle MD(f,x)(u)\ :=\ \lim _{r\to 0}{\frac {d_{X}(f(x+ru);f(x))}{r}}\,,}$

for a vector if this limit exists. The function is then called the metric differential of at that point . ${\displaystyle u\in \mathbb {R} ^{n}}$ ${\displaystyle MD(f,x)(.)}$ ${\displaystyle f}$ ${\displaystyle x}$

There is so hot at this point in the direction metric differentiable . Is even a function quite so hot in any metric differentiable . ${\displaystyle MD(f,x)(u)}$ ${\displaystyle f}$ ${\displaystyle u}$ ${\displaystyle MD(f,x)(.)}$ ${\displaystyle \mathbb {R} ^{n}}$ ${\displaystyle f}$ ${\displaystyle x}$

properties

Relation to continuity

As one can expect with a concept of differentiability , the following sentence applies.

If metrically differentiable at this point , then there is also continuous as a mapping between metric spaces. ${\displaystyle f}$ ${\displaystyle x\in \mathbb {R} ^{n}}$ ${\displaystyle f}$

Generalization of the Fréchet differential

If one understands the in a natural way (through the Euclidean norm ) as a standardized space and if the metric is also induced by a norm , then it becomes a function between standardized spaces and can thus be checked for Fréchet differentiability . In this case the following applies: ${\displaystyle \mathbb {R} ^{n}}$ ${\displaystyle d_{X}}$ ${\displaystyle \|.\|_{X}}$ ${\displaystyle f\colon (\mathbb {R} ^{n},\|.\|_{2})\to (X,\|.\|_{X})}$

If at one point Fréchet is differentiable with the differential , then it is also metrically differentiable and it continues to apply to everyone . ${\displaystyle f}$ ${\displaystyle x\in \mathbb {R} ^{n}}$ ${\displaystyle Df(x)}$ ${\displaystyle f}$ ${\displaystyle MD(f,x)(u)=\|Df(x)(u)\|_{X}}$ ${\displaystyle u\in \mathbb {R} ^{n}}$

It should be noted that the requirement is not a real restriction, because according to Kunugui's theorem , every metric space can be isometrically embedded in a Banach space. ${\displaystyle d_{X}}$

Generalization of Rademacher's theorem

If Lipschitz is continuous , the function can be differentiated metrically almost everywhere . ${\displaystyle f}$

This means that the points in which it is not differentiable form a zero set (with respect to the Hausdorff measure).

Semi-standard property

Be back Lipschitz continuous, then for almost every imaging one seminorm on . ${\displaystyle f}$ ${\displaystyle x\in \mathbb {R} ^{n}}$ ${\displaystyle MD(f,x)(.)}$ ${\displaystyle \mathbb {R} ^{n}}$

In this case it can also be shown:

For any true: . ${\displaystyle y;z\in \mathbb {R} ^{n}}$ ${\displaystyle \ d_{X}(f(y),f(z))-MD(f,x)(y-z)\in o(d(y,x)+d(x,z))}$

That is, in a - possibly very small - neighborhood of is the best isometric approximation for . In this case denote the usual Euclidian metric on ; for the use of the "small-o-notation" see also: Landau symbols ${\displaystyle x\in \mathbb {R} ^{n}}$ ${\displaystyle MD(f,x)(.)}$ ${\displaystyle f}$ ${\displaystyle d}$ ${\displaystyle \mathbb {R} ^{n}}$

There are now reversed for a - now not necessary Lipschitz - function and a place a semi-norm with the property: so has to be the same and is metric differentiable at this point. ${\displaystyle f\colon \mathbb {R} ^{n}\to (X,d_{X})}$ ${\displaystyle x\in \mathbb {R} ^{n}}$ ${\displaystyle s}$ ${\displaystyle d_{X}(f(y),f(z))-s(y-z)\in o(d(y,x)+d(x,z))}$ ${\displaystyle s}$ ${\displaystyle MD(f,x)}$ ${\displaystyle f}$

Individual evidence

1. ↑ Bernd Kirchheim: Rectifiable metric spaces: Local structure and regularity of the Hausdorff measure . Quoted from: Proceedings of the American Mathematical Society: Volume 121, Number 1, May 1994. Retrieved June 12, 2012