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 $mathbb{R}^k$ ($kleq n$). For $p>0$, we say $x$ is $d$-projectionally less than or equal to $y$ with respect to $p$-norm if $sum_{i=1}^kvert x_ivert^p$ is less than or equal to $ sum_{i=1}^kvert y_ivert^p$, for every $dleq kleq n$. For a relation $sim$ on a set $X$, we say a map $f:X rightarrow X$ is a preserver of that relation, if $xsim y$ implies $f(x)sim f(y)$, for every $x,yin X$. All the linear maps that preserve $d$-projectional equality and inequality are characterized in this paper.