DETERMINATION OF A CONVEX BODY FROM MINKOWSKI SUMS OF ITS PROJECTIONS

For a convex body K in Rd and 1 ⩽ K ⩽ d − 1, let PK (K) be the Minkowski sum (average) of all orthogonal projections of K onto k-dimensional subspaces of Rd. It is Known that the operator Pk is injective if k⩾d/2, k=3 for all d, and if k = 2, d ≠ 14. It is shown that P2k (K) determines a convex body K among all centrally symmetric convex bodies and P2k+1(K) determines a convex body K among all bodies of constant width. Corresponding stability results are also given. Furthermore, it is shown that any convex body K is determined by the two sets Pk (K) and Pk′ (K) if 1 < k < k′. Concerning the range of Pk, 1 ⩽ k ⩽ d−2, it is shown that its closure (in the Hausdorff-metric) does not contain any polytopes other than singletons.