  We initiate a study of the Euclidean Distance Degree in the context of sparse
polynomials. Specifically, we consider a hypersurface f=0 defined by a
polynomial f that is general given its support, such that the support contains
the origin. We show that the Euclidean Distance Degree of f=0 equals the mixed
volume of the Newton polytopes of the associated Lagrange multiplier equations.
We discuss the implication of our result for computational complexity and give
a formula for the Euclidean distance degree when the Newton polytope is a
rectangular parallelepiped.
