Given a vector function $Q$:$\mathbb{R}^3 \rightarrow \mathbb{R}^3 $. $S$ is a closed convex set and $\hat{n}$ is the unit vector normal to the surface $\partial S$ pointing outwards. $Q$ is not constant and is non-vanishing in $S$ and $P=\{p\in \partial S:Q=\alpha\hat{n} , \alpha \in \mathbb{R}^+\}$.
Under what conditions does $P$ has only one element ?
you can impose any conditions you need e.g. smoothness, analyticity , compactness or strict convexity .