Minimax of the angles in a plane configuration of points (Q1912647)

In Am. J. Math. 61, 912-922 (1939; Zbl 0022.19405) \textit{L. M. Blumenthal} has formulated the problem of finding the largest number \(\alpha(N)\) such that any plane configuration of \(N\) \((\geq 3)\) points contains three points determining an angle \(\pi \geq \beta \geq \alpha(N)\). Some results and conjectures were given by \textit{G. Szekeres} [Am. J. Math. 63, 208-210 (1941; Zbl 0024.13202)], \textit{P. Erdös} and \textit{G. Szekeres} [Ann. Univ. Sci. Budapest. Rolando Eötvös, Sect. Math. 3-4, 53-61 (1961; Zbl 0103.15502)] and \textit{B. Sendov} [C. R. Acad. Bulg. Sci. 45, No. 12, 17-20 (1992; Zbl 0794.52005)]. The main result in this paper is: \[ \alpha(N) = \left\{1 - {2\over 2n+1}\right\} \pi \text{ for all } 2^n < N \leq 2^n + 2^{n-2},\;n \geq 2 \] and \[ \alpha(N) = \left\{1 - {1\over n +1}\right\} \pi \text{ for all } 2^n + 2^{n-2} < N \geq 2^{n+1},\;n \geq 2. \] (Line \(1_{10}\) contains an erratum.) Along with the well-known values \(\alpha(3) = 1/3 \cdot \pi\), \(\alpha(4) = 1/2 \cdot \pi\) and \(\alpha(5) = 3/5 \cdot \pi\) now a completely solution of the problem was succeeded. For the proof is given a lot of lemmas and ideas (generalized plane configuration of points (GC), plane configuration of circles, generalized angle, perfect circle, perfect GC and others).