A uniform estimate of the relative projection constant

TL;DR

This paper establishes a quantitative lower bound greater than 1 for the relative projection constant in certain normed spaces, advancing understanding of projection behaviors in finite-dimensional Banach spaces.

Contribution

It provides a new uniform lower bound for the relative projection constant in subspaces of l_{2p}^m spaces, addressing a problem posed in 1986.

Findings

Existence of n-dimensional spaces with projection constants exceeding a specific bound

Quantitative lower bound greater than 1 for the relative projection constant

Progress on a problem posed by Bosznay and Garay in 1986

Abstract

The main goal of the paper is to provide a quantitative lower bound greater than $1$ for the relative projection constant $\lambda(Y, X)$, where $X$ is a subspace of $\ell_{2p}^m$ space and $Y \subset X$ is an arbitrary hyperplane. As a consequence, we establish that for every integer $n \geq 4$ there exists an $n$-dimensional normed space $X$ such that for an every hyperplane $Y$ and every projection $P:X \to Y$ the inequality $||P|| > 1 + \left (8 \left ( n + 3 \right )^{5} \right )^{-30(n+3)^2}$ holds. This gives a non-trivial lower bound in a variation of problem proposed by Bosznay and Garay in $1986$.