projection inequalities and their linear preservers

This paper introduces an inequality on vectors in r^n which compares vectors in r^n based on the p-norm of their projections on r^k (k ≤ n). for p > 0, we say x is d-projectionally less than or equal to y with respect to p-norm if σ^k_i =1 |yi|^p is less than or equal to σ^k_i =1 |yi|^p for every d ≤ k ≤ n. for a relation ~ on a set x, we say a map f : x → x is a preserver of that relation, if x ~ y implies f (x) ~ f (y) for every x, y є x. all the linear maps that preserve d-projectional equality and inequality are characterized in this paper.