Rectangle minimum

Usefull inequality: $\sum_{i=1}^{n} \sqrt{x_i^2 + y_i^2} \geqslant \sqrt{(\sum_{i=1}^{n} x_i)^2+(\sum_{i=1}^{n} y_i)^2}$

Lets mark segments of rectangle with $a$ and $b$ , and smaller segments by $a_1$ and $b_1$ , $a_2$ and $b_2$ , $a_3$ and $b_3$ , $a_4$ and $b_4$ , counterclockwise from bottom right vertex. Then, our S is, by Pitagora, $\sqrt{b_1^2 + a_2^2} + \sqrt{b_2^2 + a_3^2} + \sqrt{b_3^2 + a_4^2} + \sqrt{b_4^2 + a_1^2}$ . By usefull inequality: $\sqrt{b_1^2 + a_2^2} + \sqrt{a_3^2 + b_2^2} + \sqrt{b_3^2 + a_4^2} + \sqrt{a_1^2 + b_4^2} \geqslant \sqrt{(b_1+a_3+b_3+a_1)^2+(a_2+b_2+a_4+b_4)^2} = \sqrt{(2a)^2+(2b)^2} = 2\sqrt{a^2+b^2} = 10$

Example that 10 is possible is to choose midpoints of segments of rectangle, or to put two points into one vertex and other two into opposite vertex.