A symplectic capacity is a nonnegative extended-real invariant measuring the size of a symplectic manifold
under symplectic embeddings. For open
sets 
with the standard symplectic form, a normalized
capacity 
satisfies monotonicity under embeddings preserving
that symplectic form, the scaling law 
for 
, and
 |
where 
is the open unit ball in 
(Hofer and Zehnder 1994, pp. 51-67).
The Gromov width is an example. For a smooth convex body, the Ekeland-Hofer-Zehnder capacity 
is the minimum positive action of a closed characteristic
on its boundary. A characteristic is a curve tangent
to the kernel of the restricted symplectic
form, and its action is the line integral of
along the curve. This capacity extends continuously to nonsmooth convex
bodies (Haim-Kislev and Ostrover 2026).
