A uniform estimate of the relative projection constant

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 $n\ge 4$ 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 $1986$.

16 pages