Volume estimates in spaces of homogeneous polynomials
The authors consider a symmetric Banach sequence space \(X\) which is either 2-convex and has nontrivial concavity or 2-concave and has nontrivial convexity, and use \(X_n\) to denote the \(n\)-dimensional slice of \(X\) spanned by the first \(n\) basis vectors. They obtain asymptotic upper and lower bounds for the volume of the unit ball of the spaces of \(m\)-homogeneous polynomials, \({\mathcal P}(^mX_n)\), and \(m\)-homogeneous nuclear polynomials, \({\mathcal P}_N(^mX_n)\), in terms of the fundamental function of the space. This allows them to give asymptotic upper and lower bounds for the volume of the unit ball of both the full and symmetric spaces of injective and projective \(m\)-tensors on \(X_n\) in terms of the volume of the unit ball of \(X_n\). Using these results, estimates are then obtained for the volume ratio (the ratio of the volume of the ball to the volume of its ellipsoid of maximal volume) of the unit balls and the projective constants of spaces of injective and projective tensors on \(X_n\) and for the Banach--Mazur distance from the spaces of injective and projective tensors on \(X_n\) to the Hilbert spaces of the corresponding dimension. Specific and elegant formulae for all these isometric invariants are obtained in the case in which \(X\) is \(\ell_p\) with \(1\leq p\leq\infty\).
scientific article; zbMATH DE number 5533157
scientific article