Hyperplane projections of the unit ball of \(\ell_{p}^{n}\)
Inspired by various results on hyperplane sections of special unit, balls of normed linear spaces, the authors study extreme values of the volume of orthogonal hyperplane projections of unit balls of \(\ell^n_p\), i.e. of the form \[ B^n_p=\{x\in R^n:\sum^n_{i=1}|x_i|^p\leq 1\}\quad\text{with }1\leq p <\infty. \] They succeed in bringing the knowledge on this roughly to the same level as it is for sections, but they do not use the duality method. Also they show the following: If \(1\leq p\leq q =+\infty\) and \(H\) is a hyperplane of \(\mathbb{R}^n\), then \[ \frac{\text{vol}[\pi_H(B^n_q)]}{\text{vol}(B^{n-1}_q)}\geq \frac{\text{vol}[\pi_H(B^n_p)]}{\text{vol}(B^{n-1}_p)} , \] where \(\pi_H\) denotes the orthogonal projection onto \(H\).
scientific article; zbMATH DE number 1747938