A generalization of the Löwner-John´s ellipsoid theorem

We provide the following generalization of the Löwner-John´s ellipsoid theorem. Given a (non necessarily convex) compact set K ⊂ ℝⁿ and an even integer d ∈ ℕ, there is a unique homogeneous polynomial g of degree d such that K ⊂ G := {x : g(x) ≤ 1} and G has minimum volume among all such sets. The symmetric case of the Lowner-John theorem is a particular case when d = 2, and importantly, we neither require the set K nor the sublevel set G to be convex. We also provide a numerical scheme to approximate the optimal value and the unique optimal solution as closely as desired.