  We prove discrete analogs of four-vertex type theorems of spherical curves,
which imply corresponding results for space polygons. The smooth theory goes
back to the work of Beniamino Segre and, more recently, by Mohammad Ghomi, and
consists of theorems that state, for a given closed spherical curve, nontrivial
lower bounds on the number of spherical inflections plus the number of self
and/or antipodal intersections counted with multiplicity. We study these
concepts and results adapted to the case of spherical polygons and prove, using
only discrete tools, the corresponding theorems.
