A uniform estimate of the relative projection constant

The main goal of the paper is to provide a quantitative lower bound greater than 1 for the relative projection constant λ(Y,X), where X is a subspace of ℓ2pm space and Y⊂X is an arbitrary hyperplane. As a consequence, we establish that for every integer n≥4 there exists an n-dimensional normed space X such that for an every hyperplane Y and every projection P:X→Y the inequality ‖P‖>1+8n+35−30(n+3)2 holds. This gives a non-trivial lower bound in a variation of problem proposed by Bosznay and Garay in 1986.