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Minkowski sums of projections of convex bodies

Part of: General convexity

Published online by Cambridge University Press:  26 February 2010

Paul Goodey
Paul Goodey
Affiliation:
Department of Mathematics, University of Oklahoma, Norman, Oklahoma 73019, U.S.A.
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Abstract

If K is a convex body in d and 1≤k≤d − 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 k≤d/2 and that P2 is injective in all dimensions except d = 14, where it is not injective.

MSC classification

Secondary: 52A20: Convex sets in $n$ dimensions (including convex hypersurfaces)
Type
Research Article
Information
Mathematika , Volume 45 , Issue 2 , December 1998 , pp. 253 - 268
Copyright
Copyright © University College London 1998

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