Abstract
The aim of this note is to investigate the behaviour of exterior and interior parallels at distanceε to a given convex body of constant width in the unitn-sphere. The boundaries of these bodies are 2-transnormal topological (n−1)-spheres. The range ofε is determined where these transnormal topological (n−1)-spheres bound convex bodies of constant width as well. In particular this leads to the following main result: LetC be a body of constant width (π/2)+x ε ((π/2), π) in the unit sphere Sn. ThenC contains a body C of constant width (π/2)−x such thatC is the closedx-neighborhood of C. Moreover, the bodyC is smooth.
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Dedicated to Helmut Mäurer on the occasion of his 60th birthday
Supported by a Canadian NSERC Grant
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Dekster, B.V., Wegner, B. Constant width and transnormality in spheres. J Geom 56, 25–33 (1996). https://doi.org/10.1007/BF01222680
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DOI: https://doi.org/10.1007/BF01222680