Around Borsuk's Hypothesis

摘要

In the present work we will deal with one of the central problems of combinatorial geometry, namely, the problem posed by K. Borsuk in 1933. Consider an arbitrary bounded non-one-point set Ω is contained in R~n and try to represent it in the form Ω = Ω_1∪···∪Ω_f, where each Ω_i has diameter strictly smaller than the diameter of Ω. Define f(Ω) as a minimal number f for which the mentioned representation exists. In other words, we intend to partition the set fixed by us into a minimal number of parts of smaller diameter. Put f(n) = max f(Ω). Here, the maximum is taken over all Ω is contained in R~n possessing the properties indicated above. It is clear that the quantity f(n) is well-defined, although, generally speaking, it may turn out to be equal to infinity. As a matter of fact, f(n) is the minimal number of parts of smaller diameter into which an arbitrary bounded non-one-point set is partitioned in the Euclidean space of dimension n.