Global minima of multivariate constrained linear function

Question

I have a function of form $ax+by+cz$ where $a$, $b$, $c$ are real numbers. Also $x, y, z$ are greater than equal to $0$ and $x+y+z$ less than equal to $C$ (a constant). What's the global minimum of this function ($ax+by+cz$)?

Answer 1

This is a linear functional on the convex set $C= \operatorname{co} \{0, Ce_1, C e_2, C e_3 \}$. The extrema will occur at an extreme point, hence the global minimum is $\min (0, aC ,bC,cC)$.