Allometry constants of finite-dimensional spaces: theory and computations

Abstract

We describe the computations of some intrinsic constants associated to an n-dimensional normed space ${\mathcal{V}}$, namely the N-th “allometry” constants

$$\kappa_\infty^N(\mathcal{V}) := \inf \{\|T\| \cdot \|T'\|, \quad T\,:\,\ell_\infty^N \to \mathcal{V}, \; \; T': \mathcal{V} \to \ell_\infty^N, \; \;TT'={\rm Id}_{\mathcal{V}}\}.$$

These are related to Banach–Mazur distances and to several types of projection constants. We also present the results of our computations for some low-dimensional spaces such as sequence spaces, polynomial spaces, and polygonal spaces. An eye is kept on the optimal operators T and T′, or equivalently, in the case N = n, on the best conditioned bases. In particular, we uncover that the best conditioned bases of quadratic polynomials are not symmetric, and that the Lagrange bases at equidistant nodes are best conditioned in the spaces of trigonometric polynomials of degree at most one and two.