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On the X-ray Number of Almost Smooth Convex Bodies and of Convex Bodies of Constant Width

Published:2009-09-01
Printed: Sep 2009
• K. Bezdek
• Gy. Kiss
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Abstract

The X-ray numbers of some classes of convex bodies are investigated. In particular, we give a proof of the X-ray Conjecture as well as of the Illumination Conjecture for almost smooth convex bodies of any dimension and for convex bodies of constant width of dimensions $3$, $4$, $5$ and $6$.
 Keywords: almost smooth convex body, convex body of constant width, weakly neighbourly antipodal convex polytope, Illumination Conjecture, X-ray number, X-ray Conjecture
 MSC Classifications: 52A20 - Convex sets in $n$ dimensions (including convex hypersurfaces) [See also 53A07, 53C45] 52A37 - Other problems of combinatorial convexity 52C17 - Packing and covering in $n$ dimensions [See also 05B40, 11H31] 52C35 - Arrangements of points, flats, hyperplanes [See also 32S22]