HYPERPLANES OF FINITE-DIMENSIONAL NORMED SPACES WITH THE MAXIMAL RELATIVE PROJECTION CONSTANT

摘要

The relative projection constant lambda(Y, X) of normed spaces Y subset of X is lambda(Y, X) = inf{parallel to P parallel to : P is an element of P (X, Y)}, where P (X, Y) denotes the set of all continuous projections from X onto Y. By the well-known result of Bohnenblust, for every n-dimensional normed space X and a subspace Y subset of X of codimension one, lambda(Y, X) <= 2 - 2. The main goal of the paper is to study the equality case in the theorem of Bohnenblust. We establish an equivalent condition for the equality lambda(Y, X) = 2 - 2 and present several applications. We prove that every three-dimensional space has a subspace with the projection constant less than 4/3 - 0.0007. This gives a nontrivial upper bound in the problem posed by Bosznay and Garay. In the general case, we give an upper bound for the number of (n - 1)-dimensional subspaces with the maximal relative projection constant in terms of the facets of the unit ball of X. As a consequence, every n-dimensional normed space X has an (n - 1)-dimensional subspace Y with lambda(Y, X) < 2 - 2. This contrasts with the separable case in which it is possible that every hyperplane has a maximal possible projection constant.