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.

翻译：我们考虑从有限个支撑函数评估中重构具有固定面方向的多面体的任务。我们证明，对于固定的单纯形法向扇，最小二乘估计由凸二次规划给出。我们研究了该解集的几何结构，并给出了在此情况下重构唯一性的组合刻画。我们提出了一种算法，在温和假设下，随着含噪支撑函数评估数量的增加，该算法收敛到未知输入形状。我们还讨论了在去除法向扇限制时我们结果的局限性。