Beiträge zur Algebra und Geometrie Contributions to Algebra and Geometry Volume 41 (2000), No. 2, 425-426.

On a Covering Problem in the Plane

Bernulf Weißbach

Fakultät für Mathematik, Institut für Algebra und Geometrie Otto-von-Guericke-Universität Magdeburg Postschließfach 4120, D-39016 Magdeburg

Abstract. We make a remark to a covering problem stated by V. Boltyanski, H. Martini and P. S. Soltan.

At the end of their interesting book ,,Excursions into combinatorial geometry” [1], V. Boltyanski, H. Martini and P. S. Soltan present a list of problems. The latest is the following:

Problem. Let Q be a class of figures in the plane ℝ^{2}. We say that a convex figure W ⊂ ℝ^{2} is a covering set for the class Q if any figure M from the class Q can be covered by a congruent copy of W. If no convex figure W′ ⊂ W is a covering set for the class Q, then we say that W is a minimal, convex covering set for the class Q.

We consider the following classes of figures in the plane: Q_{1} is the class of all subsets of diameter 1; Q_{2} is the class of all compact, convex subsets of diameter 1; Q_{3} is the class of all figures of constant width 1; Q_{4} is the class of all Reuleaux polygons of width 1; Q_{5} is the class of all Reuleaux pentagons of width 1.

It is easy to show that the classes Q_{1}, Q_{2}, Q_{3}, Q_{4} are equivalent in the following sense: any minimal, convex covering set for one of these classes is at the same time a minimal, convex covering set for each other class. The problem is to decide whether also the class Q_{5} is equivalent (in this sense) to the previous four classes.

It is very easy to show, that the answer of this question as it is formulated is “no!”. The following theorem holds:

Theorem. There are minimal covering sets for all planar sets of constant width δ, except for closed Reuleaux triangles.

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This condition is simply to state by looking at some well-known covering sets. For instance let D_{1} be J. Pál’s hexagon and let D_{2} be the cover which H. G. Eggleston described in 1963. (This is the union of a circle and a Reuleaux triangle of diameter δ arranged in such a way, that two corners of the Reuleaux triangle are situated on the periphery of the circle [2].)

In D_{1} a Reuleaux triangle can be embedded in two positions only and in D_{2} it can be embeded even in only one way (see the figure).

If we denote the set of the corners of D_{i} by C_{i}, then D_{i}\C_{i} can not cover a closed Reuleaux triangle of the given breadth. On the other hand one can cover each planar set of constant width δ different from the Reuleaux triangles with the sets D_{i}\C_{i}. Only the Reuleaux triangles have corners with a projection cone of the measure \frac{2π}{3}. Clearly, by Zorn’s lemma it follows, that D_{i}\C_{i} must contain a minimal set with the same covering properties.

These remarks show that the problem must be reformulated as a covering problem for open sets of constant width.

References

[1] Boltyanski, V.; Martini, H.; Soltan, P. S.: Excursions into combinatorial geometry. Springer Verlag Berlin Heidelberg New York 1997.

[2] Eggleston, H. G.: Minimal universal covers in E^{n}. Israel J. Math. 1 (1963), 149–155.

Received July 29, 1999