On a characterization of the support functions (Q2366672)

The author gives the following characterization of the support functions in \(\mathbb{R}^ n\): A function \(f:S^{n-1}\to\mathbb{R}\) is the support function of a bounded subset of \(\mathbb{R}^ n\) if and only if (a) \(f\) is lower semicontinuous and bounded and (b) for every \(a\in\mathbb{R}\), if there exists \(u\in S^{n-1}\) such that \(g_ a(u):=f(u)-\langle a,u\rangle < 0\), then \(\text{Argmin }g_ a\) contains exactly one element.