"Hoffer length" measures the lengths of paths in the group Ham of Hamiltonian diffeomorphisms. A Hamiltonian circle action, as a loop in Ham, is a geodesic. If the moment map attains its maximum or minimum at an isolated fixed point with isotropy weights not all equal to plus/minus one, then this loop can be deformed into a loop in Ham of shorter Hofer length.
We give lower bounds for the possible amount of shortening and for the index (number of independent shortening directions). We also obtain a shortening result when the maximum/minimum is not isolated. These results are a completion of old joint work with Jennifer Slimowitz.