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Inequality between sides and diagonals of a space n-gon and its integral analog. (English) Zbl 0722.52006

The main result is: Let \({\mathcal A}=A_ 0,A_ 1,...,A_{n-1}\) be a closed space n-gon in \(E^ N\), let \(A_{n+k}=A_ k\) for all \(k=0,1,2,... \). Then for all \(p=0,1,...,n-1\), \[ \sum^{n-1}_{\nu =0}| A_{\nu}A_{\nu +p}|^ 2\leq [(\sin p(\pi /n))/(\sin (\pi /n))]^ 2\cdot \sum^{n-1}_{\nu =0}| A_{\nu}A_{\nu +1}|^ 2. \] For \(p=2,3,...,n-2\), equality is attained iff \({\mathcal A}\) is a plane affine-regular n-gon or, for \(N=1\), its 1-dimensional projection. (For \(p=0,1,n-1\) equality occurs always).

MSC:

52A40 Inequalities and extremum problems involving convexity in convex geometry
26D10 Inequalities involving derivatives and differential and integral operators
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