Large equilateral triangles inscribed in the unit disk of a Minkowski plane
In the Euclidean plane, both ``unit circle'' and ``equilateral triangle'' are well-understood concepts, and all equilateral triangles inscribed in the unit circle are congruent. Both concepts may be generalized to an arbitrary normed plane (the ``unit disc'' is a centrally symmetric convex body), but the situation is more complicated; the edge length of an equilateral triangle inscribed in the unit disc may vary with its orientation. For any plane with unit disc \(D\), let \(L(D)\) be the maximum edge length of any equilateral triangle inscribed in \(D\). It is known [\textit{M. Lassak}, Adv. Geom. 3, No. 1, 45--51 (2003; Zbl 1026.52004)] that \(L(D)\geq 1+ {1\over 3}\sqrt{2}\), for every normed plane. It is conjectured [\textit{P. G. Doyle}, \textit{J. C. Lagarias} and \textit{D. Randall}, Discrete Comput. Geom. 8, No. 2, 171--189 (1992; Zbl 0756.52016) and \textit{M. Lassak}, Colloq. Math. Soc. János Bolyai 63, 245--247 (1994; Zbl 0822.52001)] that this bound can be increased to \(1+{1\over 2}\sqrt{2}\), which is achieved for the regular octagon. If the norm is permitted to be asymmetric, we have the further generalization of a Minkowski plane. The unit disc is now an arbitrary convex body \(C\) containing the origin; and equilateral triangles must be defined carefully, taking the orientation of the edges into account. A triangle which is equilateral when traversed in the clockwise direction need not be equilateral counterclockwise. For some Minkowski planes, line segments with certain orientations may not be edges of any equilateral triangle whatever; and it is not immediately obvious that any of the triangles inscribed in the unit disc of a Minkowski plane need be equilateral. Nonetheless, the authors show that there is always in fact such a triangle, and \(L(C)\geq 1.25\). They conjecture that this bound can be increased to \(1.5\), and give an example (an off-center square) that shows that the true bound can be no larger. A final result concerns Minkowski planes for which the unit circle is a translate of the Euclidean circle. For such circles, it is shown that there is always an inscribed triangle with edge length at least \(\sqrt{3}\), as in the Euclidean case. (Reviewer's note: It would be natural to wonder whether \(L(D)\leq L(D+ x)\) for all centrally symmetric \(D\). However, as \(L(C)= 1.5\) was achieved for a translate of a centrally symmetric body, this cannot be so.)
scientific article; zbMATH DE number 2100239