Hofer metric from the spatial supremum norm (${\rho }_{\infty }(\varphi ,\psi )={\inf }_{{\varphi }_H^1={\varphi }^{-1}\psi }{\int }_0^1\Vert H_t{\Vert }_{\infty }dt$)

Use compactly supported Hamiltonians on open manifolds and zero-mean Hamiltonians on closed manifolds. Because their values straddle zero, $\Vert H_t{\Vert }_{\infty }\le \max H_t-\min H_t\le 2\Vert H_t{\Vert }_{\infty }$. Thus ${\rho }_{\infty }\le d_H\le 2{\rho }_{\infty }$. These are two equivalent, but not identically normalized, conventions for Hofer's metric. A statement about zero distance or zero displacement energy is independent of this choice.