- Let $D\subset \mathbb{R}^{n}$ a closed and convex set, then prove that for each $x \in D$ and $d \in \mathbb{R}^{n}$ the function
\begin{equation*} \varphi_{1}: \mathbb{R}_{+}\to \mathbb{R}, \hspace{3mm} \varphi_{1}(t)=\|P_{D}(x+td)-x\|, \end{equation*} is a monotone non-decreasing function.
- Prove that \begin{equation*} \varphi_{2}: (\mathbb{R}_{+}\setminus \{0\})\to \mathbb{R}, \hspace{3mm} \varphi_{2}(t)=\frac{\varphi_{1}(t)}{t}, \end{equation*} is a monotone non-increasing function.
I'm able to prove the first item and I'm trying to prove that the function $-\varphi_{2}(t)$ is a increasing function. My attempt: Let $\beta \geq \alpha >0$, then there exists $t \in (0,1]$ such that $\alpha = t\beta= (1-t)0+t\beta$. Here I'm stuck since I want to use that $-\varphi_{1}(t)$ is a convex function, but I can't prove this afirmation. This work or are there other form?
Thanks!
