Sum of convex functions has a minimum

Aug 2005

I have a small problem that has been hunting me for some time. Actually it seems rather simple, but I can't come up with a reasonable solution. Maybe somebody here has an idea.

Assume you have two convex functions f and g on some finite dimensional vector space. Both functions are non negative and there exist x0 and x1 such that f(x0)=0 and g(x1)=0. (These points are not necessarily unique.)

Question: Does the function f+g always has a minimiser?

In 1D this is (more or less) clear. If I'm not mistaken, then there is always at least one minimiser between x0 and x1.

Unfortunately I can't find a proof for this statement in higher dimensions.
So far I have been unable to formulate a formal proof.

The functions are continuous, but not necessarily differentiable. Thus I can't work with gradients and hessians. Maybe subgradients could be used. I haven't checked that. The theorem of Weierstrass fails also, since the level sets may be unbounded. My most promising approach so far was by assuming that there is no minimiser and considering a minimising sequence that convergences towards infinity. Unfortunately I couldn't arrive at a contradiction. The best I could achieve was f(x0)<=f(x0)...

Thanks for any ideas and suggestions,