Finding optimal shadows of polytopes

This paper deals with the problem of projecting polytopes in finite-dimensional Euclidean spaces on subspaces of given dimension so as to maximize or minimize the volume of the projection. As to the computational complexity of the underlying decision problems we show that maximizing the volume of the orthogonal projection on hyperplanes is already ℕℙ-hard for simplices. For minimization, the problem is easy for simplices but ℕℙ-hard for bipyramids over parallelotopes. Similar results are given for projections on lower-dimensional subspaces. Several other related ℕℙ-hardness results are also proved including one for inradius computation of zonotopes and another for a location problem. On the positive side, we present various polynomial-time approximation algorithms. In particular, we give a randomized algorithm for maximizing orthogonal projections of H-polytopes in ℝⁿ on hyperplanes with an error bound of essentially O(√n/log n).