  In this paper, we consider generalized moment maps for Hamiltonian actions on
$H$-twisted generalized complex manifolds introduced by Lin and Tolman
\cite{Lin}. The main purpose of this paper is to show convexity and
connectedness properties for generalized moment maps. We study Hamiltonian
torus actions on compact $H$-twisted generalized complex manifolds and prove
that all components of the generalized moment map are Bott-Morse functions.
Based on this, we shall show that the generalized moment maps have a convex
image and connected fibers. Furthermore, by applying the arguments of Lerman,
Meinrenken, Tolman, and Woodward \cite{Ler2} we extend our results to the case
of Hamiltonian actions of general compact Lie groups on $H$-twisted generalized
complex orbifolds.
