Projection Inequalities and Their Linear Preservers

This paper introduces an inequality on vectors in \mathbb{R}^n which compares vectors in \mathbb{R}^n based on the p-norm of their projections on \mathbb{R}^k (k\leq n). For p>۰, we say x is d-projectionally less than or equal to y with respect to p-norm if \sum_{i=۱}^k\vert x_i\vert^p is less than or equal to \sum_{i=۱}^k\vert y_i\vert^p, for every d\leq k\leq n. For a relation \sim on a set X, we say a map f:X \rightarrow X is a preserver of that relation, if x\sim y implies f(x)\sim f(y), for every x,y\in X. All the linear maps that preserve d-projectional equality and inequality are characterized in this paper.