Euclidean Distance Degree and Mixed Volume

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.