The solution of a convex optimization problem is unique

Question

There is a statement in a thesis I am reading,

The solution of a convex optimization problem is unique, and the global and the local minima are essentially the same.

Is there a proof for it? Why does it hold?

Answer 1

The statement is not true! For example, the convex problem: $$\min_{x} f(x) ,$$ such that $f(x)$ is given by

$$f(x) = \begin{cases} 0, & \mbox{if } -1 < x < 1 \\ \lvert x\rvert -1 , & \mbox{ otherwise} \end{cases}$$

has infinitely many local/global minima.