Hyperplanes of finite-dimensional normed spaces with the maximal relative projection constant

TL;DR

This paper investigates the conditions under which hyperplanes in finite-dimensional normed spaces attain the maximal relative projection constant, providing new bounds and characterizations that deepen understanding of projection behavior.

Contribution

It establishes an equivalent condition for maximal projection constants, offers bounds for subspaces in three-dimensional spaces, and relates the number of such subspaces to the facets of the unit ball.

Findings

Characterization of equality cases for Bohnenblust's projection constant bound

Existence of subspaces with projection constant less than 4/3 in 3D spaces

Upper bounds on the number of hyperplanes with maximal projection constant

Abstract

The \emph{relative projection constant} $\lambda(Y, X)$ of normed spaces $Y \subset X$ is defined as $\lambda(Y, X) = \inf \{ ||P|| : P \in \mathcal{P}(X, Y) \}$, where $\mathcal{P}(X, Y)$ 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 its subspace $Y$ of codimension $1$ the inequality $\lambda(Y, X) \leq 2 - \frac{2}{n}$ holds. 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 $\lambda(Y, X) = 2 - \frac{2}{n}$ and present several applications. We prove that every three-dimensional space has a subspace with the projection constant less than $\frac{4}{3} - 0.0007$. This gives a non-trivial 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 $\lambda(Y, X) < 2-\frac{2}{n}$. This contrasts with the seperable case in which it is possible that every hyperplane has a maximal possible projection constant.