Show that the function defined by $f(t)=|P_{D}(x+td)-x|$ is nondecreasing, where $D$ is closed convex, $x\in D$, $t\geq 0$, $d\in \mathbb{R}^{n}$ and $P_{D}$ is projection onto D.
I tried to solve this question in a lot of ways, for example, if we use the polarization identity, we get $\frac{1}{2}(|P_{D}(x+td)-x|^{2}-|P_{D}(x+sd)-x|^{2})= \\ \langle P_{D}(x+td)-x+P_{D}(x+sd)-x,P_{D}(x+td)-x-(P_{D}(x+sd)-x)\rangle$ where $\langle,\rangle$ stands for the usual scalar product on $\mathbb R^{n}$. I tried to explore this equality, but i got nothing.
Here is some inequalities that maybe can help: if D is closed and convex then: $
I appreciate some help, this is a problem from my homework. Thanks