Convex set: 2D representation of $\|\textbf{x} - \Pi \textbf{x}\| = \min_{\textbf{y} \in \textbf{C}}\|\textbf{x}-\textbf{y}\|$

Question

Metric projection (MP) and separation theorems

Let $\textbf{C}$ be a closed convex subset of a Hilbert space $\textbf{V}$.

For any $\textbf{x} \in \textbf{V}$, there exists a unique $\Pi \textbf{x} \in \textbf{C}$ such that $$ \|\textbf{x} - \Pi \textbf{x}\| = \min_{\textbf{y} \in \textbf{C} \quad(1)}\|\textbf{x}-\textbf{y}\| \quad (2) $$

Here are my questions:

• Does this 2D drawing corresponds to the statement
• If yes can we say in that case that this implies that $ \Pi \textbf{x} = \textbf{y}$? (3)