A Remark On Hofer-like Geometry

We show that Banyaga’s Hofer-like norm, a generalization of the Hofer norm coincides with the classical Hofer norm when restricted to Hamiltonian diffeomorphisms on compact symplectic manifolds. This result proves a conjecture of Banyaga and fills the gap between Hofer and Hofer-like geometries: the refined Hofer-like structure degenerates to standard Hofer geometry within the Hamiltonian subgroup. This equality allows the straightforward extension of essential results from Hofer geometry to the Hofer-like setting.