  We introduce an extension of the standard Local-to-Global Principle used in
the proof of the convexity theorems for the momentum map to handle closed maps
that take values in a length metric space. This extension is used to study the
convexity properties of the cylinder valued momentum map introduced by
Condevaux, Dazord, and Molino and allows us to obtain the most general
convexity statement available in the literature for momentum maps associated to
a symplectic Lie group action.
