  Let $(M,\omega)$ be a symplectic manifold and $U\subseteq M$ an open subset.
We study the natural inclusion of the compactly supported Hamiltonian group of
$U$ in the compactly supported Hamiltonian group of $M$. The main result is an
upper bound for this map in terms of the Hofer norms for $U$ and $M$.
  Applications are upper bounds on the asymptotic Hofer-Lipschitz constant and
the relative Hofer diameter of $U$. The first bound is often sharp and the
second one is often sharp up to a factor of 2.
