Let K:=x:g(x)≤1\documentclass[12pt]{minimal}
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\begin{document}$$\mathbf{K }:=\left\{ \mathbf{x }: g(\mathbf{x })\le 1\right\} $$\end{document} be the compact (and not necessarily convex) sub-level set of some homogeneous polynomial g\documentclass[12pt]{minimal}
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\begin{document}$$g$$\end{document}. Assume that the only knowledge about K\documentclass[12pt]{minimal}
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\begin{document}$$\mathbf{K }$$\end{document} is the degree of g\documentclass[12pt]{minimal}
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\begin{document}$$g$$\end{document} as well as the moments of the Lebesgue measure on K\documentclass[12pt]{minimal}
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\begin{document}$$\mathbf{K }$$\end{document} up to order 2d\documentclass[12pt]{minimal}
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\begin{document}$$2d$$\end{document}. Then the vector of coefficients of g\documentclass[12pt]{minimal}
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\begin{document}$$g$$\end{document} is the solution of a simple linear system whose associated matrix is nonsingular. In other words, the moments up to order 2d\documentclass[12pt]{minimal}
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\begin{document}$$2d$$\end{document} of the Lebesgue measure on K\documentclass[12pt]{minimal}
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\begin{document}$$\mathbf{K }$$\end{document} encode all information on the homogeneous polynomial g\documentclass[12pt]{minimal}
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\begin{document}$$g$$\end{document} that defines K\documentclass[12pt]{minimal}
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\begin{document}$$\mathbf{K }$$\end{document} (in fact, only moments of order d\documentclass[12pt]{minimal}
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\begin{document}$$d$$\end{document} and 2d\documentclass[12pt]{minimal}
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\begin{document}$$2d$$\end{document} are needed).
