---
tag: 8KCB5KJKNT
kind: concept
name: Symplectic capacity
msc: [53D35, 57R17]
sources:
  - url: https://en.wikipedia.org/wiki/Non-squeezing_theorem
---

## Summary

A symplectic capacity is a number attached to symplectic manifolds, or to subsets of $\R^{2n}$, that measures their size.
It is compatible with symplectic embeddings.
It cannot grow under symplectic embeddings and is normalized so that a ball and a cylinder of the same radius have the same capacity.
The existence of capacities is equivalent in strength to [[Gromov's non-squeezing theorem|RDX55HQ3XT]], and capacities are a main tool of quantitative symplectic topology.

## Explanations

### L1 human

Volume is a natural size for a region of space, but in symplectic geometry it misses much.
A thin infinite cylinder has infinite volume, yet a large ball cannot be fitted into it symplectically.
A capacity is a size that sees this: it behaves like an area, measured in a two-dimensional symplectic direction.
The simplest one is the Gromov width, the area $\pi r^2$ of the largest ball of radius $r$ that fits into a set by a symplectic embedding.