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Let's say we have some polytope $P$ in 3-space (which is not necessarily convex) as well as some number of points on its surface, $(g_1, ..., g_N)$. We are provided no information about the coordinates for any given point, $g_i$, or information about the underlying geometry of $P$. However, we are able to draw a finite number of spheres of radius $(r_1, ..., r_M)$, with some $g_i$ their centerpoints, and find the volume of the intersection of each sphere with the polytope $P$. Here, all $r_i$ are less than at least the largest cross-sectional dimension of $P$.

As a function of the number of coordinates $N$, to what extent can we learn about the geometry of $P$ using this information?

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Perhaps this is a route to explore: Interpret your volumes as solid angles, and from them obtain an estimate of the curvature at your points $g_i$. Then apply the remarkable theorem of Gluck, Krigelman, and Singer:

"The converse to the Gauss-Bonnet Theorem in PL," J. Diff. Geom, 9(4): 601-616, 1974.

which says—essentially—that given curvatures of points on a manifold $M$ that satisfy the obvious necessary conditions, there exists a PL Riemannian metric on $M$ which realizes those curvatures at the specified points, and is flat elsewhere.

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