One class of extremal geometric constants and their applications (Q909743)

Let X be a linear normed space over R and \(\bar w=(w_ 1,...,w_ n)\in R^ n\); \(r,n\in N\); \(r\leq n\). The authors investigate extremal geometric constants \[ \delta_ r(\bar w;X)=\inf_{x_ 1,...,x_ n\in X;\| x_ j\| \geq 1;j=1,...,r}\max_{\pi \in S_ n}\| \sum^{n}_{j=1}w_{\pi (j)}x_ j\|, \] where \(S_ n\) is the set of all permutations of indices 1,2,...,n. This paper contains a generalization of theorems of the authors' previous results [Set theory, Foundations of Mathematics, Proc. Symp. Belgrade 1977, Zb. Rad., Nova Ser. 2(10), 129-137 (1977; Zbl 0497.05024); Mat. Zametki 29, 691-709 (1981; Zbl 0482.05013), translation in Math. Notes 29, 352-361 (1981); Izv. Akad. Nauk SSSR, Ser. Mat. 46, 535-568 (1982; Zbl 0513.60013)] and their applications in matrix theory.