  LVM and LVMB manifolds are a large family of examples of non k\"ahler
manifolds. For instance, Hopf manifolds and Calabi-Eckmann manifolds can be
seen as LVMB manifolds. The LVM manifolds have a very natural action of the
real torus and the quotient of this action is a simple polytope. This quotient
allows us to relate closely LVM manifolds to the moment-angle manifolds studied
(for example) by Buchstaber and Panov. The aim of this paper is to generalize
the polytopes associated to LVM manifolds to the LVMB case and study its
properties. In particular, we show that it belongs to a very large class of
simplicial spheres. Moreover, we show that for every sphere belonging to this
class, we can construct a LMVB manifold whose associated sphere is the given
sphere. We use this latter result to show that many moment-angle complexes can
be endowed with a complex structure (up to product with circles).
