Unique reconstruction of convex bodies by projection data via refinements of the Brunn-Minkowski inequality

Algorithmic problems in geometry often become tractable with the assumption of convexity. Optimization, volume computation, geometric learning and finding the centroid are all examples of problems that are significantly easier for convex sets. One of the most powerful results in Convex Geometry is the so-called Brunn-Minkowski theorem. It plays a very important role as much in the theoretical framework of geometric inequalities as well as in applied contexts. For instance in crystallography, where it is used to show the mathematical interpretation of the Gibbs-Curie principle (the equilibrium shape of the crystal minimizes the surface energy among all sets of the same volume); or in order to ensure the concavity of the objective function in some optimization problems relative to algorithms for reconstructing a polytope P from surface area measures data. Brunn-Minkowski inequality establishes that the n-th root of the volume of two n-dimensional convex bodies K,E is a concave function, and assuming a common hyperplane projection of K and E (or a common maximal volume section through parallel hyperplanes to a given one), it was proved that the volume itself is concave, i.e., vol(tK+(1-t)E)>tvol(K)+(1-t)vol(E). In this talk we will show, on the one hand, that under the sole assumption that K and E have an equal volume projection (or a common maximal volume section), if the above linear inequality holds with equality for just one value of t in (0,1), then K may be specifically recovered via K=L+E, with L being a segment. We will also discuss that this extra assumption is needed in order to obtain such a characterization. On the other hand, we will show that the expected result of concavity for the k-th root of the volume, when a common projection onto an (n-k)-plane (or a common maximal volume (n-k)-section) is assumed, is not true, by explicitly giving (a family of) convex bodies providing a counterexample for this statement. Nevertheless, other Brunn-Minkowski type inequalities can be derived under this hypothesis. Furthermore, the proofs of these results are constructive in the sense that such bodies can be completely reconstructed by means of the appropriate geometric data in each case