## 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

### Keywords:

Wirtinger’s inequality; Fourier polynomials; space n-gon
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