Minkowski sums of projections of convex bodies

If K is a convex body in E-d and 1 less than or equal to k less than or equ al to d - 1, we define P-k(K) to be the Minkowski sum or Minkowski average of all the projections of K onto k-dimensional subspaces of E-d. The operat or Pd-1 was first introduced by Schneider, who showed that, if Pd-1(K) = cK , then K is a ball. More recently, Spriestersbach showed that, if Pd-1(K) = Pd-1(M) then K = M. In addition, she gave stability versions of this resul t and Schneider's. We will describe further injectivity results for the ope rators P-k. In particular, we will show that P-k is injective if k greater than or equal to d/2 and that P-2 is injective in all dimensions except d = 14, where it is not injective.