Ratios of volumes and factorization through \(\ell_ \infty\)
We develop a method effective in the computation of projection constants \(\lambda(X)\) for \(n\)-dimensional normed, as well as quasi-normed spaces \(X\). We show that for every centrally symmetric compact body \(K\) in \(\mathbb{R}^n\) there exist a parallelotope \(P\) and a zonoid \(Z\) such that \(Z\subset K\subset P\), and \((\text{vol}_n(P)/\text{vol}_n(Z))^{1/n}\leq \lambda(X)\), where \(\lambda(X)\) is the projection constant of the quasi-normed space \(X= (\mathbb{R}^n, |\cdot|)\) having \(K\) as its unit ball. Thus, it follows easily that the projection constant of the quasi-normed space \(\ell^n_p\) \((0< p< 1)\) is equivalent to \(n^{1/p- 1/2}\). This improves the estimate due to T. Peck. The method also yields easily the projection constants of the quasi-normed Schatten classes \(s^n_p\) \((0< p< 1)\), consisting of operators on \(\ell^n_2\), and other examples, including classical examples of normed spaces. In addition, we establish relations between various other known parameters associated with the geometry of normed spaces, such as volume-ratio, the \(\ell_\infty\) norm of operators between quasi-normed spaces, and others. We prove that if \(X\) is a normed space of dimension \(n\) then the unit balls \(B\) of \(X\) and \(B^*\) of \(X^*\), contain zonoids \(Z_1\) and \(Z_2\) respectively, such that \(({\text{vol}_n(B) \text{vol}_n(B^*)\over \text{vol}_n(Z_1) \text{vol}_n(Z_2)})^{1/n}\leq c\text{ gl}_2(X)\), where \(c\) is an absolute constant and \(\text{gl}_2(X)\) is the Gordon-Lewis parameter of \(X\). We show also that if \(K\) is the convex hull of \(m\) points in \(\mathbb{R}^n\), \(n\leq m\), then there is a simplex \(\Delta\) inside \(K\) satisfying the inequality \((\text{vol}_n(K))^{1/n}\leq c\sqrt{\ln{2m\over n}} (\text{vol}_n(\Delta))^{1/n}\).
scientific article; zbMATH DE number 878730