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  • Cited by 3
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    This article has been cited by the following publications. This list is generated based on data provided by CrossRef.

    Goodey, Paul and Weil, Wolfgang 2006. Average section functions for star-shaped sets. Advances in Applied Mathematics, Vol. 36, Issue. 1, p. 70.

    Goodey, Paul and Weil, Wolfgang 2006. Directed Projection Functions of Convex Bodies. Monatshefte für Mathematik, Vol. 149, Issue. 1, p. 43.

    Kiderlen, Markus 2002. Determination of the mean normal measure from isotropic means of flat sections. Advances in Applied Probability, Vol. 34, Issue. 03, p. 505.


Minkowski sums of projections of convex bodies

  • Paul Goodey (a1)
  • DOI:
  • Published online: 01 February 2010

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
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