In section 2.6.3 of "Convex Optimization" by Boyd and Vandenberghe (p. 55) it is stated that
If $\lambda\succ_{K^*}0$ and $x$ minimizes $\lambda^T z$ over $z\in S$, then $x$ is minimal.
This is complemented a bit later with the assertion
Provided the set $S$ is convex, we can say that for any minimal element $x$ there exists a non-zero $\lambda\succeq_{K^*}0$ such that $x$ minimizes $\lambda^T z$ over $z\in S$.
In my application, $K$ is the non negative orthant, $S$ is indeed convex and $\lambda^T z$ can be shown to have a unique minimum over $z\in S$ for any $\lambda\succeq 0$.
Can I say that $x$ is minimal if and only if $x$ minimizes $\lambda^T z$ over $z\in S$ for some non zero $\lambda\succeq 0$?