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?