Learning Polytopes with Fixed Facet Directions

We consider the task of reconstructing polytopes with fixed facet directions from finitely many support function evaluations. We show that for a fixed simplicial normal fan the least-squares estimate is given by a convex quadratic program. We study the geometry of the solution set and give a combinatorial characterization for the uniqueness of the reconstruction in this case. We provide an algorithm that, under mild assumptions, converges to the unknown input shape as the number of noisy support function evaluations increases. We also discuss limitations of our results if the restriction on the normal fan is removed.