Mathematika

Research Article

Minkowski sums of projections of convex bodies

Paul Goodeya1

a1 Department of Mathematics, University of Oklahoma, Norman, Oklahoma 73019, U.S.A.

Abstract

If K is a convex body in d and 1≤kd − 1, we define Pk(K) to be the Minkowski sum or Minkowski average of all the projections of K onto k-dimensional subspaces of d. The operator 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) = cK then K = M. In addition, she gave stability versions of this result and Schneider's. We will describe further injectivity results for the operators Pk. In particular, we will show that Pk is injective if kd/2 and that P2 is injective in all dimensions except d = 14, where it is not injective.

(Received July 04 1997)

Key Words:

  • 52A20: CONVEX AND DISCRETE GEOMETRY; General convexity; convex sets in n dimensions.