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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

Stability results for first order projection bodies


Authors: P. R. Goodey and H. Groemer
Journal: Proc. Amer. Math. Soc. 109 (1990), 1103-1114
MSC: Primary 52A40
MathSciNet review: 1015678
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Abstract: The motivation for this work comes from a result of Minkowski. He showed that if a three-dimensional convex body has the property that all its projections have the same perimeter, then the original body has constant width. Our objective was to extend this to a stability result and not to restrict ourselves to dimension three. The result we obtained shows that if two centrally symmetric bodies have projections which all have approximately the same mean width, then the two bodies are approximately the same up to translation. This is, in effect, a continuity result for the inverse of the spherical Radon transform. It is closely related to recent three-dimensional results of Campi and to work of Bourgain and Lindenstrauss, who consider the volumes of projections rather than their mean widths. The techniques we employ are drawn from the theory of spherical harmonics and from the theory of mixed volumes.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1990-1015678-5
PII: S 0002-9939(1990)1015678-5
Article copyright: © Copyright 1990 American Mathematical Society