Isoperimetric inequalites and areas of projections in \(\mathbb{R}^ n\)
Let \({\mathcal C}\) be the family of all smooth closed curves of given length \(L\) in \(\mathbb{R}^ n\), and, for each \(C \in {\mathcal C}\), let \[ A = {1 \over 2} \sum^ n_{j = 1} \sum^ n_{ k = 1} \varepsilon_{jk} A_{jk} \] be the sum of the projected areas \(A_{jk}\) onto all planes \(Ox_ j x_ k\) with an assigned index \(\varepsilon_{jk} \in \{- 1,0,1\}\) such that \(\varepsilon_{kj} = - \varepsilon_{jk}\). The authors determine the greatest value of \(| A |/L^ 2\) over \({\mathcal C}\) and they also find a curve \(C \in {\mathcal C}\) for which this value is attained.
scientific article; zbMATH DE number 575894
scientific article