Projectional inequalities and their linear preservers

This paper introduces an inequality on vectors in Rn which compares vectors in Rn based on the p-norm of their projections on Rk (k n). For p > 0, we say x is d-projectionally less than or equal to y with respect to p-norm if Σki =1 jxijp is less than or equal to Σki =1 jyijp 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 2 X. All the linear maps that preserve d-projectional equality and inequality are characterized in this paper.