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The inverse moment problem for convex polytopes. (English) Zbl 1285.68198
Summary: We present a general and novel approach for the reconstruction of any convex $$d$$-dimensional polytope $$P$$, assuming knowledge of finitely many of its integral moments. In particular, we show that the vertices of an $$N$$-vertex convex polytope in $$\mathbb R^{d}$$ can be reconstructed from the knowledge of $$O(DN)$$ axial moments (w.r.t. to an unknown polynomial measure of degree $$D$$), in $$d+1$$ distinct directions in general position. Our approach is based on the collection of moment formulas due to Brion, Lawrence, Khovanskii-Pukhlikov, and Barvinok that arise in the discrete geometry of polytopes, combined with what is variously known as Prony’s method, or the Vandermonde factorization of finite rank Hankel matrices.

##### MSC:
 68U05 Computer graphics; computational geometry (digital and algorithmic aspects) 52B05 Combinatorial properties of polytopes and polyhedra (number of faces, shortest paths, etc.) 52A22 Random convex sets and integral geometry (aspects of convex geometry)
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