 Zero displacement energy for a compact set in a hyperplane

Use the standard symplectic form ${\sum }_{j}d{q}_j\wedge d{p}_j$ and convention ${\iota }_{X_H}\omega =-dH$. The Hamiltonian $H=\delta q_n$ translates $p_n$ by $\delta$. A fixed compactly supported cutoff equal to one near the set swept for $|\delta |\le 1$ leaves that translation unchanged on the original compact set. Its supremum norm is at most $C|\delta |$, so the translation displaces the set at arbitrarily small cost. This proves zero Hamiltonian displacement energy for either Hofer normalization.