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A generalization of Löwner-John's ellipsoid theorem
International audienceWe address the following generalization of the Lowner-John ellipsoid problem. Given a (non necessarily convex) compact set and an even integer , find an homogeneous polynomial of degree such that and has minimum volume among all such sets. We show that is a convex optimization problem even if neither nor are convex! We next show that has a unique optimal solution and a characterization with at most contacts points in is also provided. This is the analogue for of the Lowner-John's theorem in the quadratic case , but importantly, we neither require the set nor the sublevel set to be convex. More generally, there is also an homogeneous polynomial of even degree and a point such that and 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