Why is a local min also a global min for convex functions?

Question

As the title states,

for an unconstrained minimizaton problem, of a convex function, why is it that the local minimum is also the global solution?

Answer 1

Assuming unconstrained optimization, let $x^*$ be a local minima of $f$. Then $f(x^*)\le f(x)\ \forall x$ in some neighborhood of $x^*$. Now, let $y$ be a point on a line joining $x^*$ and $z$ such that $y$ belongs to this neighborhood. Then, $y=\alpha x^*+(1-\alpha)z$ for some $\alpha\in [0,1]$. Then, $f(x^*)\le f(y)=f(\alpha x^*+(1-\alpha)z)\le \alpha f(x^*)+(1-\alpha)f(z)\implies f(x^*)\le f(z)$ for $\alpha\ne 1$, if $\alpha=1$, then $y=z$ and $z$ is in the neighborhood mentioned before so by hypothesis, $f(x^*)\le f(z)$. Thus $x^*$ is a global minimizer.