On a covering problem in the plane
Referring to a problem (p. 392: Problem 28) posed in the book ``Excursions into combinatorial Geometry'', Berlin/Heidelberg/New York (1997; Zbl 0877.52001), by \textit{V. Boltyanski, H. Martini} and \textit{P. S. Soltan}, the author proves the Theorem: There are (sets which are) minimal covering sets for all planar sets of constant width \(d\), except for closed Reuleaux triangles. To prove this, denote by \(D\) a Reuleaux triangle of width \(d\) and consider either a regular hexagon \(D_1\) whose opposite sides have the distance \(d\), or a convex domain \(D_2\) the boundary of which consists of two arcs of the boundary of \(D\) and a semicircle (H. G. Eggleston, 1963). \(D\) can be embedded in \(D_i\) in \(3-i\) positions (\(i=1,2\)). If one denotes the set of the corners of \(D_i\) by \(C_i\), then \(D_i\setminus C_i\) cannot cover a closed Reuleaux triangle \(D\), but -- on the other hand -- one can cover each planar set of constant width \(d\) different from the Reuleaux triangles with the sets \(D_i\setminus C_i\) (only the Reuleaux triangles have corners with a projection cone of the measure \(2\pi/3\)). By Zorn's lemma it follows that \(D_i\setminus C_i\) must contain a minimal set with the same covering properties. In consequence of the theorem above, the questions of ``Problem 28'' (among them: ``Is any minimal, convex covering set for the class of all figures of constant width 1 also a minimal, convex covering set for the class of all Reuleaux polygons of width 1, and vice versa?'') must be reformulated as covering problems for open sets (of constant width).
scientific article; zbMATH DE number 1505615