We address the following generalization P\documentclass[12pt]{minimal}
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\begin{document}$$\mathbf {P}$$\end{document} of the Löwner-John ellipsoid problem. Given a (non necessarily convex) compact set K⊂Rn\documentclass[12pt]{minimal}
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\begin{document}$$\mathbf {K}\subset \mathbb {R}^n$$\end{document} and an even integer d∈N\documentclass[12pt]{minimal}
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\begin{document}$$d\in \mathbb {N}$$\end{document}, find an homogeneous polynomial g\documentclass[12pt]{minimal}
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\begin{document}$$g$$\end{document} of degree d\documentclass[12pt]{minimal}
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\begin{document}$$d$$\end{document} such that K⊂G:={x:g(x)≤1}\documentclass[12pt]{minimal}
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\begin{document}$$\mathbf {K}\subset \mathbf {G}:=\{\mathbf {x}:g(\mathbf {x})\le 1\}$$\end{document} and G\documentclass[12pt]{minimal}
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\begin{document}$$\mathbf {G}$$\end{document} has minimum volume among all such sets. We show that P\documentclass[12pt]{minimal}
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\begin{document}$$\mathbf {P}$$\end{document} is a convex optimization problem even if neither K\documentclass[12pt]{minimal}
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\begin{document}$$\mathbf {K}$$\end{document} nor G\documentclass[12pt]{minimal}
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\begin{document}$$\mathbf {G}$$\end{document} are convex! We next show that P\documentclass[12pt]{minimal}
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\begin{document}$$\mathbf {P}$$\end{document} has a unique optimal solution and a characterization with at most n+d-1d\documentclass[12pt]{minimal}
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\begin{document}$${n+d-1\atopwithdelims ()d}$$\end{document} contacts points in K∩G\documentclass[12pt]{minimal}
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\begin{document}$$\mathbf {K}\cap \mathbf {G}$$\end{document} is also provided. This is the analogue for d>2\documentclass[12pt]{minimal}
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\begin{document}$$d>2$$\end{document} of Löwner-John’s theorem in the quadratic case d=2\documentclass[12pt]{minimal}
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\begin{document}$$d=2$$\end{document}, but importantly, we neither require the set K\documentclass[12pt]{minimal}
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\begin{document}$$\mathbf {K}$$\end{document} nor the sublevel set G\documentclass[12pt]{minimal}
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\begin{document}$$\mathbf {G}$$\end{document} to be convex. More generally, there is also an homogeneous polynomial g\documentclass[12pt]{minimal}
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\begin{document}$$g$$\end{document} of even degree d\documentclass[12pt]{minimal}
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\begin{document}$$d$$\end{document} and a point a∈Rn\documentclass[12pt]{minimal}
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\begin{document}$$\mathbf {a}\in \mathbb {R}^n$$\end{document} such that K⊂Ga:={x:g(x-a)≤1}\documentclass[12pt]{minimal}
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\begin{document}$$\mathbf {K}\subset \mathbf {G}_\mathbf {a}:=\{\mathbf {x}:g(\mathbf {x}-\mathbf {a})\le 1\}$$\end{document} and Ga\documentclass[12pt]{minimal}
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\begin{document}$$\mathbf {G}_\mathbf {a}$$\end{document} 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.
