Suppose that $f(x):R^p \rightarrow R$ is a convex function with global minimum, say 0.
Let $C=(x: f(x)=0)$, i.e. the set of the global minimum. Suppose that there exist at least one point $y$ such that $f(y) = 0$,
It is easy to see that $C$ is convex subset.
Let $a_{\lambda}$ such that $f(a_{\lambda})$ approach 0 as $\lambda$ approach 0 and let $a_{\lambda}^1$ be the closest point to $a_{\lambda}$ in $C$.
Prove that $|a_{\lambda}^1-a_{\lambda}|$ converge to 0, i.e. $a_{\lambda}$ approaches $C$ as $\lambda$ approach 0.