Abstract:
Let $f(d)$ be the minimum number of parts of smaller diameter into which one can partition an arbitrary bounded subset of $d$-dimensional Euclidean space $\mathbb R^d$. In 1933, Borsuk conjectured that $f(d)=d+1$. Recent results of Kahn–Kalai, Nilli, and the present author demonstrate that the class of integral polytopes is one of the most important classes having a direct connection with Borsuk's conjecture and problems close to it.
In the present paper, with the use of the methods of the set-covering problem new upper bounds are obtained for the minimum number of parts of smaller diameter into which each $d$-dimensional $(0,1)$-polytope or cross-polytope can be partitioned. These bounds are substantially better than the author's similar former results as well as all previously known bounds for $f(d)$.
In addition, $(0,1)$-polytopes and cross-polytopes in small dimensions are studied in this paper.
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