Some inequalities and properties concerning chordal semi-polygons (Q2720291)

Let \(P=A_1A_2\dots A_n\) be an ordinary polygon in the Euclidean plane with vertices \(A_1,A_2,\dots,A_n\). A polygon is called chordal if each of its vertices lies on one common circle \({\mathcal C}\). NEWLINENEWLINENEWLINEThe theorems and corollaries in the paper under review contain a wealth of inequalities involving (oriented) angles like \(\beta_i = \angle CA_iA_{i+1}\) and \(\varphi_i = \angle A_iCA_{i+1}\), where \(C\) is the center of \({\mathcal C}\). Obviously, \(-\pi/2 < \beta_i < \pi/2\) and \(-\pi \leq \varphi_i \leq \pi\) holds. For chordal polygons \(A_1A_2\dots A_n\) the author proves among many other results NEWLINE\[NEWLINE\sum_{i=1}^n \alpha_i = 2 \sum_{i=1}^n \beta_i \tag{1}NEWLINE\]NEWLINE and NEWLINE\[NEWLINE\sum_{i=1}^n \beta_i > 0 \quad \text{for \(n\) odd}.\tag{2}NEWLINE\]NEWLINE Most results, however, address \(k\)-chordal polygons which are defined as follows. A chordal polygon \(P = A_1A_2\dots A_n\) is of the first kind, if there is a point \(O\) in the interior of the circle \({\mathcal C}\) such that all angles \(\psi^O_i =\angle A_iOA_{i+1}\) have the same sign. If for such a polygon \(P\) there exists some point \(M\) in the interior of \({\mathcal C}\) such that \(|\sum_{i=1}^n \psi^M_i|= 2k\pi\), where \(k \in {\mathbb N}\) is maximal, then \(P\) is called a \(k\)-chordal polygon. The author proves for \(k\)-chordal polygons results like: NEWLINENEWLINENEWLINE(1) \(\sum_{i=1}^n \cos^p \beta_i > n (2k/n)^p\) for any integer \(p > 0\), and NEWLINENEWLINENEWLINE(2) if \(a_i\) is the length of edge \(A_iA_{i+1}\) and if \(a_1\) is the maximum of these lengths, then NEWLINE\[NEWLINE(\sum_{i=1}^n a_i)/a_1+(\sum_{i=1}^n a_i^3) / 6 a_1^3+\dots+(\sum_{i=1}^n a_i^{2k+1}) / (2k+1)! a_1^{2k+1} + \dots > k\pi.NEWLINE\]NEWLINE This inequality also characterizes \(k\)-chordal polygons.