---
tag: Y7P66D41Z8
kind: concept
name: Flat norm
msc: [49Q15, 28A75]
sources:
  - url: https://en.wikipedia.org/wiki/Current_(mathematics)
---

## Summary

The flat norm measures the distance between two [[currents|360H7MD9RD]] by the least total size of a decomposition.
The difference is split into a small piece plus the boundary of a small filling.
Two surfaces are close in the flat norm when they differ by a small area or bound a thin region.
This holds even if their masses differ.
Convergence in the flat norm is the natural convergence of generalised surfaces, and the [[Federer–Fleming compactness theorem|TPXYARH912]] is stated with it.
It also gives the distance used in the comparison and approximation of [[integral currents|EG4AJVEFT2]].

## Explanations

### L1 human

Two nearby points, one counted positively and one negatively, look very different to the mass, which counts each point once and gives a total of $2$.
But they are the two ends of a short segment, so they are almost the same once one allows filling in.
The flat norm captures this: the difference is the boundary of a segment of length $\varepsilon$, so its flat norm is at most $\varepsilon$.
In the same way, a wiggly curve close to a straight one is near it in the flat norm, because the two curves bound a thin region.