### 1999-15

#### Sets with a large Borsuk number

Series: 1999-15, Preprints

MSC:
52A20 Convex sets in $n$ dimensions (including convex hypersurfaces)

Abstract:
We construct sets in Euclidean spaces of dimension
d= $(4m-2 \over 2)$, where m is a power of a prime,
with the property that they can only be coverd with a
large number of sets having smaller diameter.
Thereby we generalize a result of A. M. Raigorodskii and,
in addition, we prove that there exists a counterexample to
the so called 'Borsuk-conjecture' already in dimension
\$(34 \over 2) - 1 = 560.

Keywords:
Borsuk's problem