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 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, 52A37, 52C17, 52C35 show english descriptions Convex sets in $n$ dimensions (including convex hypersurfaces) [See also 53A07, 53C45]
Other problems of combinatorial convexity
Packing and covering in $n$ dimensions [See also 05B40, 11H31]
Arrangements of points, flats, hyperplanes [See also 32S22] 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]

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