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For the spatial supremum-norm convention, a normalized Hamiltonian function gives the path lengthOn an open manifold use compactly supported Hamiltonians; on a closed manifold use zero-mean Hamiltonians, which fixes their additive time-dependent constants. The Hofer metric from the spatial supremum norm isEquivalently one minimizes this length over Hamiltonian paths joining the two maps. Inversion and conjugation preserve the generator's supremum norm, so the distance is symmetric and bi-invariant; concatenation gives the triangle inequality. Its nondegeneracy is the geometric assertion proved in Question 3, not a consequence of these formal properties alone.

Another common convention for Hofer's metric uses the oscillation length ${\ell }_{\mathrm{osc}}(H)={\int }_0^1(\max H_t-\min H_t)dt$, giving a distance $d_H$. Normalized or compactly supported Hamiltonians have values straddling zero, soThus these conventions give equivalent distances, and agree on whether an energy vanishes. We keep the same chosen convention in every energy estimate.

The Hamiltonian displacement energy of a subset $A$ isThe infimum is $+\infty$ if there is no displacing map. In the oscillation convention replace ${\rho }_{\infty }$ by $d_H$.

Choose coordinates $(q_1,\dots ,q_n,p_1,\dots ,p_n)$ with standard symplectic form ${\omega }_0={\sum }_{j}d{q}_j\wedge d{p}_j$, and regard the given sphere $S$ as lying in $\{p_n=0\}$. With ${\iota }_{X_H}{\omega }_0=-dH$, the linear Hamiltonian function $H_{\delta }=\delta q_n$ has Hamiltonian vector field $\delta {\partial }_{p_n}$ and translates in the missing coordinate.

To make this a permissible compactly supported map, take a smooth cutoff $\chi$ equal to one on an open neighbourhood of the fixed compact swept setChoose $\chi$ with compact support and let $R={\sup }_{\mathrm{supp}\chi }|q_n|<\infty$. For $0<\delta <1$ put $G_{\delta }=\delta q_n\chi$. On the whole swept region the cutoff is locally constant, so its derivatives vanish and $X_{G_{\delta }}=\delta {\partial }_{p_n}$. Thus the flow carries every $z\in S$ to $z+t\delta e_{p_n}$ for $0\le t\le 1$.

The time-one image lies in $\{p_n=\delta \}$ and is disjoint from the original sphere. Its Hofer metric from the spatial supremum norm obeysIn the oscillation convention the upper bound is $2R\delta$. Since $\delta$ can approach zero while the cutoff and $R$ stay fixed,This proves the zero displacement energy for a compact set in a hyperplane; it does not assert that any one nonidentity displacing map has zero norm.