Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Spherical coverings and X-raying convex bodies of constant width   K. Bezdek and Gy. Kiss showed that existence of origin-symmetric coverings of
unit sphere in $\mathbb{E}^n$ by at most $2^n$ congruent spherical caps with
radius not exceeding $\arccos\sqrt{\frac{n-1}{2n}}$ implies the $X$-ray
conjecture and the illumination conjecture for convex bodies of constant width
in $\mathbb{E}^n$, and constructed such coverings for $4\le n\le 6$. Here we
give such constructions with fewer than $2^n$ caps for $5\le n\le 15$.
  For the illumination number of any convex body of constant width in
$\mathbb{E}^n$, O.~Schramm proved an upper estimate with exponential growth of
order $(3/2)^{n/2}$. In particular, that estimate is less than $3\cdot 2^{n-2}$
for $n\ge 16$, confirming the above mentioned conjectures for the class of
convex bodies of constant width. Thus, our result settles the outstanding cases
$7\le n\le 15$.
  We also show how to calculate the covering radius of a given discrete point
set on the sphere efficiently on a computer.

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The categories of this paper are: hep-th
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The categories of this paper are: acc-phys
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The categories of this paper are: nucl-ex
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The categories of this paper are: quant-ph
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The categories of this paper are: hep-lat
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The categories of this paper are: dg-ga
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The categories of this paper are: econ
16
The categories of this paper are: stat
29
The categories of this paper are: funct-an
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The categories of this paper are: chao-dyn
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The categories of this paper are: q-alg
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The categories of this paper are: q-fin
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The categories of this paper are: nlin
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The categories of this paper are: cond-mat
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The categories of this paper are: bayes-an
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The categories of this paper are: physics