  A theorem of Tietze and Nakamija, from 1928, asserts that if a subset X of
R^n is closed, connected, and locally convex, then it is convex. We give an
analogous "local to global convexity" theorem when the inclusion map of X to
R^n is replaced by a map from a topological space X to R^n that satisfies
certain local properties.
  We say that a map from a topological space to R^n is convex if every two
points in the space can be connected by a path whose composition with the map
is a weakly monotone parametrization of a straight line segment. Let X be a
connected Hausdorff topological space, let T be a convex subset of R^n, and let
Psi: X \to T be a continuous proper map. Suppose that every point in X is
contained in an open set U such that the map Psi|_U: U \to Psi(U) is convex and
open. Then the map Psi: X \to \Psi(X) is convex and open. Consequently, its
image is convex and its level sets are connected.
  Our motivation comes from the Condevaux-Dazord-Molino proof of the
Atiyah-Guillemin-Sternberg convexity theorem in symplectic geometry.
