Abstract:
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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