  Each integrable lowest weight representation of a symmetrizable Kac-Moody Lie
algebra g has a crystal in the sense of Kashiwara, which describes its
combinatorial properties. For a given g, there is a limit crystal, usually
denoted by B(-\infty), which contains all the other crystals. When g is finite
dimensional, a convex polytope, called the Mirkovi\'c-Vilonen polytope, can be
associated to each element in B(-\infty). This polytope sits in the dual space
of a Cartan subalgebra of g, and its edges are parallel to the roots of g. In
this paper, we generalize this construction to the case where g is a symmetric
affine Kac-Moody algebra. The datum of the polytope must however be
complemented by partitions attached to the edges parallel to the imaginary root
\delta. We prove that these decorated polytopes are characterized by conditions
on their normal fans and on their 2-faces. In addition, we discuss how our
polytopes provide an analog of the notion of Lusztig datum for affine Kac-Moody
algebras. Our main tool is an algebro-geometric model for B(-\infty)
constructed by Lusztig and by Kashiwara and Saito, based on representations of
the completed preprojective algebra \Lambda\ of the same type as g. The
underlying polytopes in our construction are described with the help of Buan,
Iyama, Reiten and Scott's tilting theory for the category \Lambda-mod. The
partitions we need come from studying the category of semistable
\Lambda-modules of dimension-vector a multiple of \delta.
