# Separation

1. Let $S_1, S_2$ are closed convex sets such that: $S_1 \cap S_2 = \varnothing$. Is it true that $% $
2. Find a separating hyperplane between $S_1$ and $S_2$:

3. Find a supporting hyperplane for a set of $S = \left\{ x \in \mathbb{R}^2 \mid e^{x_1} \le x_2\right\}$ at the boundary point $x_0 = (0.1)$
4. Find a supporting hyperplane for the set of $S = \left\{ x \in \mathbb{R}^3 \mid x_3 \ge x_1^2 + x_2^2\right\}$ such that separates it from the point $x_0 = \left(-\frac{5}{4}, \frac{5}{16}, \frac{15}{16}\right)$
5. Приведите пример двух строго, но не сильно отделимых множеств. Двух отделимых, но не собственно отделимых множеств.
6. Illustrate the difference between the tangenti hyperplane and the separating hyperplane by considering a convex set with a non-smooth boundary.
7. Derive the equation of the supporting hyperplane of the set of $\{ x \in \mathbb{R}^3 \mid \dfrac{x_1^2}{4} + \dfrac{x_2^2}{9} + \dfrac{x_3^2}{25} \le 1 \}$ at $(-6/5, 12/5, 0), (0, 9/5, 4), (6/5, 0, -4)$ (any choice)
8. Derive the equation of the supporting hyperplane of the set of $\{ x \in \mathbb{R}^3 \mid x_3 \ge x_1^2 + x_2^2 \}$, which separates it from the points $(5/4, 5/16, 15/16), (4/3, 2/3, 13/12), (11/9, 11/27, 1)$ (any choice)
9. Find a separating hyperplane between $S_1$ and $S_2$:

10. Find a supporting hyperplane for a set $S = \left\{ x \in \mathbb{R}^3 \mid \frac{x_1^2}{4}+\frac{x_2^2}{8}+\frac{x_3^2}{25} \le 1 \right\}$ at the border point $x_0 = \left(-1, \frac{12}{5}, \frac{\sqrt{3}}{2}\right)$