There is a convex polyhedron $P \subset \mathbb{R}^{3}$ and there are its planar sections $S_{1}, \ldots, S_{n}$ througth planes $\pi_{1}, \ldots, \pi_{n}$, $S_{i} \subset \pi_{i}$. All these $S_{i}$ are planar polygons.
We don't know neither $P$ nor its sections $S_{1}, \ldots, S_{n}$.
We only know $S^{*}_{1}, \ldots, S^{*}_{n}$ that are noisy measurements of $S_{1}, \ldots, S_{n}$, $S^{*}_{i} \subset \pi_{i}$. It's suggested that for each $i = 1, \ldots, n$ both $S^{*}_{i}$ and $S_{i}$ has the same number of sides and vertices, i. e. they have the same topologycal structure.
Are there any ideas how to reconstruct the convex polyhedron $P^{*}$ from $S^{*}_{1}, \ldots, S^{*}_{n}$ which will be the best approximation of $P$ by these measurements?