Geodesic
The problems of Borsuk and Gr\"unbaum on lattice polytopes
Izvestiya. Mathematics , Tome 69 (2005) no. 3, pp. 513-537

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We study two classical problems of combinatorial geometry, the Borsuk problem on partitioning sets into parts of smaller diameter and the Grünbaum problem on covering sets by balls. We obtain new non-trivial upper bounds for the minimum number of parts of smaller diameter into which an arbitrary lattice polytope can be partitioned, as well as for the minimum number of balls of the same diameter by which any such polytope can be covered. 
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     title = {The problems of {Borsuk} and {Gr\"unbaum} on lattice polytopes},
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A. M. Raigorodskii. The problems of Borsuk and Gr\"unbaum on lattice polytopes. Izvestiya. Mathematics , Tome 69 (2005) no. 3, pp. 513-537. http://geodesic.mathdoc.fr/item/IM2_2005_69_3_a2/