Finding minimum with geometry?

Question

We have variables $x_{1},x_{2},x_{3},...,x_{n}>0$ and constants $c_{1},c_{2},c_{3},...,c_{n}>0$ where $n=2k, k\in\mathbb{N}$

Find the minimum of $\sum_{cyc}\sqrt{x^2_{1}+\left(x_{2}-c_{1}\right)^{2}}$ Express your answer with constants $c_{1},c_{2},c_{3},...,c_{n}$

Answer 1

Let $\sum\limits_{i=1}^nx_i=X$ and $\sum\limits_{i=1}^nc_i=C$.

Thus, by Minkowski (triangle inequality) and C-S we obtain: $$\sum_{i=1}^n\sqrt{x_i^2+(x_{i+1}^2-c_i)^2}\geq\sqrt{X^2+(X-C)^2}=$$ $$=\frac{1}{\sqrt2}\sqrt{(1+1)(X^2+(C-X)^2)}\geq \frac{X+C-X}{\sqrt2}=\frac{C}{\sqrt2}.$$

Answer 2

My idea:

Therefore the length of hypotenuse is $$\sqrt{(c_{1}+c_{3}+c_{5}+...+c_{n-1})^2+(c_{2}+c_{4}+c_{6}+...+c_{n})^2} \ge \dfrac{c_{1}+c_{2}+c_{3}+...+c_{n-1}+c_{n}}{\sqrt{2}}$$