A generalization of L"owner-John's ellipsoid theorem

We address the following generalization P of the Lowner-John ellipsoid problem. Given a (non necessarily convex) compact set Ksubset Rn and an even integer d, find an homogeneous polynomial g of degree d such that Ksubset G:=x:g(x)≤1 and G has minimum volume among all such sets. We show that P is a convex optimization problem even if neither K nor G are convex! We next show that P has a unique optimal solution and a characterization with at most n+d-1choose d contacts points in Kcap G is also provided. This is the analogue for dtextgreater2 of the Lowner-John's theorem in the quadratic case d=2, but importantly, we neither require the set K nor the sublevel set G to be convex. More generally, there is also an homogeneous polynomial g of even degree d and a point a∈ Rn such that Ksubset Ga:=x:g(x-a)≤1 and Ga has minimum volume among all such sets (but uniqueness is not guaranteed). Finally, we also outline a numerical scheme to approximate as closely as desired the optimal value and an optimal solution. It consists of solving a hierarchy of convex optimization problems with strictly convex objective function and Linear Matrix Inequality (LMI) constraints.