Let $p: \mathbb{R}^n \rightarrow X$ be the projection mapping onto $X \subset \mathbb{R}^n$ compact convex.
I am wondering if $$ \left( p(x) - p(y) \right)^\top z \leq \left( x -y\right)^\top z $$ for all $x,y,z \in \mathbb{R}^n$.
What I came up with is just $\left( x - p(x)\right)^\top \left( p(x) - z \right) \geq 0$ for all $x,z \in \mathbb{R}^n$ and obviously $\left\| p(x) - p(y) \right\| \leq \left\| x-y\right\|$ for all $x,y \in \mathbb{R}^n$.
No. For example, if you project to a singleton $X=\{x_0\}$ then $\left( p(x) - p(y) \right)^\top z$ is always zero, while $\left( x -y\right)^\top z$ may well be negative. One could try to salvage the inequality with absolute values: $$|\left( p(x) - p(y) \right)^\top z| \leq |\left( x -y\right)^\top z| \tag{2}$$ but this one fails as well. For example, in $\mathbb R^2$ let $X$ be the unit disk, $z = (1,0)$, $x=(1,2)$, $y=(1,1)$. The right side of (2) is zero, but the left is not.