Borsuk number for planar convex bodies. (English) Zbl 1440.52001
Let \(C\) be a bounded set in the Euclidean space \(\mathbb{R}^n\). The Borsuk number \(\alpha(C)\) of \(C\) is the minimal number of subsets with strictly smaller diameters into which \(C\) can be decomposed. The original question by Borsuk asks if \(\alpha(C)\leq n+1\) for any \(C\). The answer is positive for \(n=2\) and \(n=3\), and for any dimension when \(C\) is a smooth or centrally symmetric convex body. For the general case, the least-dimension with known negative answer is \(n=64\), but it has not been proved to be optimal. In \(\mathbb{R}^2\), \(\alpha(C)=3\) iff the completion of \(C\) by a constant width set with the same diameter is unique, and \(\alpha(C)=2\) iff there exists a segment \(s\) in \(C\) whose length is not the diameter of \(C\) and \(s\) intersects every diameter segment of \(C\).
The Introduction of this paper is very clear and interesting and includes a complete review of the historical results of this topic in the Euclidean space. The authors consider the graph \(G_C(V,E)\), where the vertices \(V\) are the end points of the diameter segments of \(C\), and \(E\) are the diameter segments of \(C\). They study Borsuk numbers for centrally symmetric planar convex bodies in Section 3, and prove that \(\alpha(C)=2\) iff \(V\neq \partial C\). Finally the following known fact is obtained: the Euclidean ball is the unique centrally symmetric planar convex body with Borsuk number equals to 3. Section 4 is devoted to general compact sets in \(\mathbb{R}^n\) (without the assumption of convexity), and they obtain that \(\alpha(C)=2\) iff \(G_C\) is bipartite and \(V\) can be decomposed into two subsets \(V_R\) and \(V_B\) such that the intersection of the closures of \(V_R\) and \(V_B\) is empty. Finally some examples and figures of planar convex sets with Borsuk numbers equal to three are presented.
B. Grünbaum [Bull. Res. Council Israel, Sect. F 7, 25–30 (1957; Zbl 0086.15202)] studied the Borsuk numbers of sets with respect to a metric distinct from the Euclidean, and determined the Borsuk numbers of sets in the plane equipped with the \(L_{\infty}\) norm. The problem was solved for arbitrary normed planes by V. G. Boltyanskij and V. P. Soltan [Math. Notes 22, 839–844 (1978; Zbl 0418.52004)]. The \(k\)-fold Borsuk number is a generalization of Borsuk’s problem [M. Hujter and Z. Lángi, Isr. J. Math. 199, Part A, 219–239 (2014; Zbl 1304.52013)]. It could be interesting to apply the theory of graph in order to improve the results with non Euclidean metrics and with the extension to \(k\)-fold Borsuk number.
The Introduction of this paper is very clear and interesting and includes a complete review of the historical results of this topic in the Euclidean space. The authors consider the graph \(G_C(V,E)\), where the vertices \(V\) are the end points of the diameter segments of \(C\), and \(E\) are the diameter segments of \(C\). They study Borsuk numbers for centrally symmetric planar convex bodies in Section 3, and prove that \(\alpha(C)=2\) iff \(V\neq \partial C\). Finally the following known fact is obtained: the Euclidean ball is the unique centrally symmetric planar convex body with Borsuk number equals to 3. Section 4 is devoted to general compact sets in \(\mathbb{R}^n\) (without the assumption of convexity), and they obtain that \(\alpha(C)=2\) iff \(G_C\) is bipartite and \(V\) can be decomposed into two subsets \(V_R\) and \(V_B\) such that the intersection of the closures of \(V_R\) and \(V_B\) is empty. Finally some examples and figures of planar convex sets with Borsuk numbers equal to three are presented.
B. Grünbaum [Bull. Res. Council Israel, Sect. F 7, 25–30 (1957; Zbl 0086.15202)] studied the Borsuk numbers of sets with respect to a metric distinct from the Euclidean, and determined the Borsuk numbers of sets in the plane equipped with the \(L_{\infty}\) norm. The problem was solved for arbitrary normed planes by V. G. Boltyanskij and V. P. Soltan [Math. Notes 22, 839–844 (1978; Zbl 0418.52004)]. The \(k\)-fold Borsuk number is a generalization of Borsuk’s problem [M. Hujter and Z. Lángi, Isr. J. Math. 199, Part A, 219–239 (2014; Zbl 1304.52013)]. It could be interesting to apply the theory of graph in order to improve the results with non Euclidean metrics and with the extension to \(k\)-fold Borsuk number.
Reviewer: Pedro Martín Jiménez (Badajoz)
MSC:
| 52A10 | Convex sets in \(2\) dimensions (including convex curves) |
References:
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