Secant varieties of toric varieties

Let XPXP be a smooth projective toric variety of dimension nn embedded in PrPr using all of the lattice points of the polytope PP. We compute the dimension and degree of the secant variety SecXP. We also give explicit formulas in dimensions 2 and 3 and obtain partial results for the projective varieties XAXA embedded using a set of lattice points A⊂P∩ZnA⊂P∩Zn containing the vertices of PP and their nearest neighbors.