# Projection

1. Let us have two different points $a, b \in \mathbb{R}^n$. Prove that the set of points which in the Euclidean norm are closer to the point $a$ than to $b$ make up a half-space. Is this true for another norm?
2. Find $\pi_S (y) = \pi$ if $S = \{x \in \mathbb{R}^n \mid \|x - x_c\| \le R \}$, $y \notin S$
3. Find $\pi_S (y) = \pi$ if $S = \{x \in \mathbb{R}^n \mid c^T x = b \}$, $y \notin S$
4. Find $\pi_S (y) = \pi$ if $S = \{x \in \mathbb{R}^n \mid Ax = b, A \in \mathbb{R}^{m \times n}, b \in \mathbb{R}^{m} \}$, $y \notin S$
5. Illustrate the geometric inequality that connects $\pi_S(y), y \notin S, x \in S$, from which it follows that $\pi_S(y)$ is a projection of the $y$ point onto a convex set of $S$.
6. For which sets does the projection of the point outside this set exist? Uniquie?
7. Find $\pi_S (y) = \pi$ if $S = \{x \in \mathbb{R}^n \mid c^T x \ge b \}$
8. Find $\pi_S (y) = \pi$ if $S = \{x \in \mathbb{R}^n \mid x = x_0 + X \alpha, X \in \mathbb{R}^{n \times m}, \alpha \in \mathbb{R}^{m}\}$, $y \in S$
9. Let $S \subseteq \mathbb{R}^n$ be a closed set, and $x \in \mathbb{R}^n$ be a point not lying in it. Show that the projection in $l_2$ norm will be unique, while in $l_\infty$ norm this statement is not valid.
10. Find the projection of the matrix $X$ on a set of matrices of rank $k, \;\;\; X \in \mathbb{R}^{m \times n}, k \leq n \leq m$. In Frobenius norm and spectral norm.
11. Find a projection of the $X$ matrix on a set of symmetrical positive semi-definite matrices of $X \in \mathbb{R}^{n \times n}$. In Frobenius norm and the scalar product associated with it.
12. Find the projection $\pi_S(y)$ of point $y$ onto the set $S = \{x_1, x_2 \in \mathbb{R}^2 \mid \mid \vert x_1\vert + \vert x_2\vert = 1 \}$ in $\| \cdot \|_1$ norm. Consider the different positions of $y$.
13. Find $\pi_S (y) = \pi$, if $S = \{x \in \mathbb{R}^n \mid \alpha_i \le x_i \le \beta_i, i = 1, \ldots, n \}$.
14. Prove that projection is a nonexpansive operator, i.e. prove, that if $S \in \mathbb{R}^{n}$ is nonempty, closed and convex set, then for any $(x_{1}, x_{2}) \in \mathbb{R}^{n} \times \mathbb{R}^{n}$