Convex function in $R^n$ - does existence of minimum in every direction imply existence of global minimum?

Question

Let $f:R^n\rightarrow R$ ($n>1$) be a convex function satisfying condition:

$\forall x\in R^n \ \exists t_0 = \arg\min\limits_{t\in R} f(tx).$

Is it true that there exists point $x_0\in R^n$ being global minimum of $f$?

EDIT: $t_0$ does not need to be unique.

Answer 1

It was quite tough to come up with this counterexample.

Take $$f(x,y) = \max( y^2 - x, -y ).$$ It is pretty easy to see that this function is convex and the existence of the "directional minimizers" is straightforward to check. However, $$f( s^2 + s, s) = \max( s^2 - s^2 - s, -s) = -s$$ shows that $f$ is not bounded from below. Hence, it does not have a global minimizer.