We introduce Fenchel-Nielsen coordinates on Teicm\"uller spaces of surfaces
of infinite type. The definition is relative to a given pair of pants
decomposition of the surface. We start by establishing conditions under which
any pair of pants decomposition on a hyperbolic surface of infinite type can be
turned into a geometric decomposition, that is, a decomposition into hyperbolic
pairs of pants. This is expressed in terms of a condition we introduce and
which we call Nielsen convexity. This condition is related to Nielsen cores of
Fuchsian groups.