  We define and solve the toric version of the symplectic ball packing problem,
in the sense of listing all 2n-dimensional symplectic-toric manifolds which
admit a perfect packing by balls embedded in a symplectic and torus equivariant
fashion.
  In order to do this we first describe a problem in geometric-combinatorics
which is equivalent to the toric symplectic ball packing problem. Then we solve
this problem using arguments from Convex Geometry and Delzant theory.
  Applications to symplectic blowing-up are also presented, and some further
questions are raised in the last section.
