A generalization of the Lowner-John’s ellipsoid theorem

We address the following generalization $P$ of the Lowner-John's ellipsoid problem. Given a (non necessarily convex) compact set $K\subset R^n$ and an even integer $d, find an homogeneous polynomial$g$of degree$d$such that$K\subset G:=\{x:g(x)\leq1\}$and$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-1\choose d}$contacts points in$K\cap G$is also provided. This is the analogue for$d>2$of the Lowner-John's theorem in the quadratic case$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\in R^n$such that$K\subset G_a:=\{x:g(x-a)\leq1\}$and$G_a\$ 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.