In my course of optimization I got the following definition of metric projections
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 $$ \|\textbf{x} - \Pi \textbf{x}\| = \min_{y \in \textbf{C}}\|\textbf{x}-\textbf{y}\| $$
- $\textbf{y}_o \in \textbf{C}$ equals $\Pi \textbf{x}$ exactly when (iff) $$ \langle \textbf{x}- \textbf{y}_o, \textbf{y} - \textbf{y}_o \rangle \leq 0 \text{ for all }\textbf{y} \in \textbf{C} $$
- If $\textbf{C}$ is a closed convex cone, then $\textbf{y}_o \in \textbf{C}$ equals $\Pi \textbf{x}$ iff $$ \langle\textbf{x} - \textbf{y}_o, \textbf{y}_o \rangle = 0 \text{ and } \langle\textbf{x} - \textbf{y}_o, \textbf{y} \rangle \leq 0 \text{ for all } \textbf{y} \in \textbf{C} $$
- If $\textbf{C}$ is a closed linear subspace \textbf{V}, then $\textbf{y}_o \in \textbf{C}$ equals $\Pi \textbf{x}$ iff $$ \langle\textbf{x} - \textbf{y}_o, \textbf{y} \rangle = 0 \text{ for all } y \in \textbf{C} $$
- For any two points $\textbf{x}_1$, $\textbf{x}_2 \in \textbf{V}$, $$ \|\Pi \textbf{x}_1 - \Pi \textbf{x}_2\| \leq \|\textbf{x}_1 - \textbf{x}_2\| $$
Here are my questions:
- what does $\Pi$ represent? is it a scalar $\in \mathbb{R}$? is it something else?
- what does the notation $\min_{y\in \textbf{C}}$ exactly mean?
- what does $y_o$ represent? is it an element at the center of a Ball?
$\Pi$, here, is supposed to be a mapping $V \rightarrow C$.
In the statements 2 to 4, $x$ is an arbitrary element of $V$ and $y_o$ an arbitrary element of $C$.
$\min_{y \in C} \|x-y\|$ is the smallest element of $\{\|x-y\|,\,y \in C\}$.