Let $C$ be a non-empty closed convex set of $\mathbb{R}^n$ and $x \in \mathbb{R}^n$. a. Show that there exists a unique point $P_C(x) \in C$ such that $\|P_C(x) - x\| = \inf_{y \in C} \|y - x\|$. b. Let $\bar{x} \in C$. Show that $\bar{x} = P_C(x)$ if and only if $$\langle x - \bar{x}, y - \bar{x} \rangle \leqslant 0 \text{ for all } y \in C.$$ Hint: one may consider the function $\psi_y : t \in \mathbb{R} \mapsto \|x - (\bar{x} + t(y - \bar{x}))\|^2$, where $y \in C$. c. Deduce that if $x, y \in \mathbb{R}^n$, then $\|P_C(y) - P_C(x)\| \leqslant \|y - x\|$.
Let $C$ be a non-empty closed convex set of $\mathbb{R}^n$ and $x \in \mathbb{R}^n$.\\
a. Show that there exists a unique point $P_C(x) \in C$ such that $\|P_C(x) - x\| = \inf_{y \in C} \|y - x\|$.\\
b. Let $\bar{x} \in C$. Show that $\bar{x} = P_C(x)$ if and only if
$$\langle x - \bar{x}, y - \bar{x} \rangle \leqslant 0 \text{ for all } y \in C.$$
Hint: one may consider the function $\psi_y : t \in \mathbb{R} \mapsto \|x - (\bar{x} + t(y - \bar{x}))\|^2$, where $y \in C$.\\
c. Deduce that if $x, y \in \mathbb{R}^n$, then $\|P_C(y) - P_C(x)\| \leqslant \|y - x\|$.