Suppose for any subspace $F$ of $\mathbb{R}^d$(with the usual Euclidean norm), $\pi_F$ denote the orthogonal projection onto $F$. Let $R$ be a rotation of $F$ and $F^\prime = RF$. Prove that $\forall p \in \mathbb{R}^d$,
$d(\pi_F(p), \pi_{F^\prime}(p)) \leq d(\pi_F(p), R\pi_F(p))$.
This relation is used in this paper, equation 4.5.
If I well understand the question the inequality is not true in general.
As suggested in my comment a simple counterexample can be done in $\mathbb{R}^2$, as we can see in this figure.
Here the line $OC$ is the subspace $F$ and the line $OD$ is the subspace $F'=R_\alpha F$. For a point $P$ we have the projections $E_1=\pi_F(P)$ and $E_2=\pi_{F'}(P)$, and the point $G=R_\alpha(E_1)$.
We can see that $d(E_1E_2)=d(E_1Q)$ if $P$ is a point of the bisector of the angle $\alpha$, otherwise we can have $d(E_1E_2)>d(E_1Q)$ or $d(E_1E_2)<d(E_1Q)$ depending on the position of $P$.