On \((k, f,l)\)-chordal polygons (Q2703568)

If \(A\) is the \(k\)-chordal polygon with sides of lengths \(a_1,\dots ,a_n\) and \(f(a_1),\dots ,f(a_n)\) are the side lengths of an \(l\)-chordal polygon, then \(A\) is called \((k,f,l)\)-chordal polygon. It is known that if \(a_1,\dots ,a_n\) are the side lengths of a \(k\)-chordal polygon, then \(\sum_{i=1}^n a_i>\gamma(k,n)a^*\), where \(a^*=\) max \(\{a_j\mid j=1,\dots ,n\}\) and \(\gamma(k,n)\geq n \sin (k\pi /n)\). NEWLINENEWLINENEWLINETheorem 2.2. Let \(a_1,\dots ,a_n\) be any given lengths with \(\sum_{i=1}^n a_i>2ka^*\). Then \(A\) is \((j,f,l)\)-chordal polygon for \(j=1,\dots ,k\) if \(f: R_+\to R_+\) is convex, monotonously increasing, \(f(0)=0\), and NEWLINE\[NEWLINE\arcsin \frac{f(2ka^*/n)}{f(a^*)}>\frac{l\pi}{n}.NEWLINE\]NEWLINE If also \(A\) is a \(k\)-chordal polygon, \(k\leq [(n-1)/2]\), then \(A\) is \((k,f,l)\)-chordal if \(f: R_+\to R_+\) is convex, monotonously nondecreasing and NEWLINE\[NEWLINE\arcsin \frac{f(2ka^*/n)}{f(a^*)}>\frac{l\pi}{n}\qquad (\text{Theorem }2.1).NEWLINE\]NEWLINE Theorem 2.3. Let \(a_1,\dots ,a_n\) be any given lengths with \(\sum_{i=1}^n f(a_i)>\gamma(k,n)a^*\), where \(k\leq [(n-1)/2]\). Assume \(\gamma(k,n)\geq n\sin (k\pi /n)\). If the positive increasing function \(f\) is star-like with respect to the origin, then there is a \((k,f,k)\)-chordal polygon whose sides have lengths \(a_1,\dots ,a_n\).