  Given a two-dimensional space endowed with a divergence function that is
convex in the first argument, continuously differentiable in the second, and
satisfies suitable regularity conditions at Voronoi vertices, we show that
orphan-freedom (the absence of disconnected Voronoi regions) is sufficient to
ensure that Voronoi edges and vertices are also connected, and that the dual is
a simple planar graph. We then prove that the straight-edge dual of an
orphan-free Voronoi diagram (with sites as the first argument of the
divergence) is always an embedded triangulation.
  Among the divergences covered by our proofs are Bregman divergences,
anisotropic divergences, as well as all distances derived from strictly convex
$\mathcal{C}^1$ norms (including the $L_p$ norms with $1< p < \infty$). While
Bregman diagrams of the {first kind} are simply affine diagrams, and their
duals ({weighted} Delaunay triangulations) are always embedded, we show that
duals of orphan-free Bregman diagrams of the \emph{second kind} are always
embedded.
