What does this function look like?

Question

I'm having a hard time visualizing this function:

$$f(x) = \min_{\alpha_1,\alpha_2,\alpha_3\in[0,1]}\{\alpha_1x, \alpha_2x,\alpha_3x\}$$

such that $\alpha_1+\alpha_2+\alpha_3=1$.

Answer 1

In the following $\alpha_k \ge 0$ and $\sum_k \alpha_k = 1$.

If $x \ge 0$, then $\alpha_k x \ge 0$ for all $k$ and choosing $\alpha = (1,0,0)$ we get $f(x) = 0$.

If $x < 0$, then $x \le \alpha_k x$ for all $k$ and choosing $\alpha = (1,0,0)$ we get $f(x) = x$.

Hence $f(x) = \min(0,x)$.