A generalization of Löwner-John’s ellipsoid theorem

Abstract

We address the following generalization $$\mathbf {P}$$ of the Löwner-John ellipsoid problem. Given a (non necessarily convex) compact set $$\mathbf {K}\subset \mathbb {R}^n$$ and an even integer $$d\in \mathbb {N}$$ , find an homogeneous polynomial $$g$$ of degree $$d$$ such that $$\mathbf {K}\subset \mathbf {G}:=\{\mathbf {x}:g(\mathbf {x})\le 1\}$$ and $$\mathbf {G}$$ has minimum volume among all such sets. We show that $$\mathbf {P}$$ is a convex optimization problem even if neither $$\mathbf {K}$$ nor $$\mathbf {G}$$ are convex! We next show that $$\mathbf {P}$$ has a unique optimal solution and a characterization with at most $${n+d-1\atopwithdelims ()d}$$ contacts points in $$\mathbf {K}\cap \mathbf {G}$$ is also provided. This is the analogue for $$d>2$$ of Löwner-John’s theorem in the quadratic case $$d=2$$ , but importantly, we neither require the set $$\mathbf {K}$$ nor the sublevel set $$\mathbf {G}$$ to be convex. More generally, there is also an homogeneous polynomial $$g$$ of even degree $$d$$ and a point $$\mathbf {a}\in \mathbb {R}^n$$ such that $$\mathbf {K}\subset \mathbf {G}_\mathbf {a}:=\{\mathbf {x}:g(\mathbf {x}-\mathbf {a})\le 1\}$$ and $$\mathbf {G}_\mathbf {a}$$ 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 constraints.