Borsuk covering and planar sets with unique completion

✍ Scribed by Krzysztof Kołodziejczyk

• Publisher: Elsevier Science
• Year: 1993
• Tongue: English
• Weight: 617 KB
• Volume: 122
• Category: Article
• ISSN: 0012-365X
• DOI: 10.1016/0012-365x(93)90298-8

✦ Synopsis

The famous problem of Borsuk, whether every bounded set in aB" can be covered by n+ 1 sets of smaller diameter, is still open for n > 4. We give an equivalent formulation of the problem. In the plane, the only sets which cannot be covered by two sets of smaller diameter are those whose completion is unique. We present a new characterization of planar sets with unique completion.

Borsuk's problem is still open, although some partial results have been obtained. For example, the best known estimates for b(W) are n+1<b(R")<min{J(1/3)(n+2)3(2+&$-1, 2"-'+l).

The upper bound is a combination of the results of Danzer [8] and Lassak [16] (see also [lS]). Moreover, the Borsuk number of different classes of sets has been