 Solution

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 $d_H$.