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Minkowski sums of projections of convex bodies

  • Paul Goodey (a1)

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.

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H. Fallert , P. Goodey and W. Weil . Spherical projections and centrally symmetric sets. Adv. in Math., 129 (1997), 301322.

C. Muller . Spherical Harmonics (Springer, Berlin, 1966).

R. Schneider . Rekonstruktion eines konvexen Korpers aus seinen Projektionen. Math. Nachr., 79 (1977), 325329.

R. Schneider and W. Weil . Zonoids and related topics. In: Convexity and its Applications P. M. Gruber and J. M. Wills , eds. (Birkhauser, Basel, 1983), 296317.

R. Vitale . Lp metrics for compact convex sets, J. Approx. Theory, 45 (1985), 280287.

W. Weil . Zonoide und verwandte Klassen konvexer Korper, Monatshefte Math., 94 (1982), 7384.

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  • ISSN: 0025-5793
  • EISSN: 2041-7942
  • URL: /core/journals/mathematika
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