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A generalization of Löwner-John’s ellipsoid theorem. (English) Zbl 1337.90049
In this remarkable work, the author considers the following geometric problem. Given a compact set $$K$$ of $${\mathbb R}^n$$ and an even integer $$d$$, find a homogeneous polynomial $$g$$ of degree $$d$$ such that its unit sublevel set $$G:=\{x \in {\mathbb R}^n : g(x) \leq 1\}$$ contains $$K$$ and has minimal volume (i.e. Lebesgue measure). This turns out to be a finite-dimensional convex optimization problem, even though $$K$$ is not assumed to be convex or connected. This follows from the observation that the volume of $$G$$ is proportional to the integral $$\int_{{\mathbb R}^n} \exp(-g(x))dx$$. Using first-order optimal conditions, the author then shows that this optimization problem has a unique optimal solution, and a tight upper bound on the number of contact points of $$K$$ and $$G$$ is also provided.
Whereas finding the optimal set $$G$$ boils down to solving a finite-dimensional convex optimization problem (whose decision variable is the homogeneous polynomial $$g$$), solving this optimization problem seems to be a challenge, the main difficulty being evaluation of integrals of exponentials of polynomials. Numerically, the optimal set $$G$$ can however be approximated as closely as desired with a hierarchy of finite-dimensional semidefinite programming problems for which efficient interior-point algorithms are available.
This work can be considered as a significant extension of the Löwner-John ellipsoid theorem stating that there exists a minimum volume ellipsoid $$G$$ containing a given convex compact set $$K$$. The extension is twofold. First, $$K$$ is not assumed to be convex or connected. Second, $$G$$ is not restricted to be the unit sublevel set of a degree $$2$$ polynomial.

##### MSC:
 90C22 Semidefinite programming 65K10 Numerical optimization and variational techniques
##### Software:
GloptiPoly; Polynomial Toolbox
Full Text:
##### References:
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