An algebra problem by Ace star

Let $a \geq b \geq c$ , then $|a-b| + |b-c| + |c-a| = 2 (a - c)$ . To maximize the value, we need to maximize $a$ and minimize $c$ ; $b$ doesn't affect the value. Let $a^2 + c^2 = k^2 \leq 1$ , then $|2ac| \leq k^2$ :

$(a - c)^2 = a^2 + c^2 - 2ac = 1 - b^2 - 2ac \leq 1 - 2ac \leq 1 + k^2 \leq 2$

So the value $2 (a - c) \leq \boxed{2 \sqrt{2}}$ . The maximum is achieved with following: $a = \frac{\sqrt{2}}{2}, b = 0, c = - \frac{\sqrt{2}}{2}$ .