Support function of a set: when we can replace the supremum with the maximum?

Consider a set $K\subseteq \mathbb{R}^d$ (not necessarily closed and/or convex). Let $k:=(k_1,...,k_i,...,k_d)$ be a generic element of $K$.

Consider the function $h_K:\mathbb{R}^d\rightarrow \mathbb{R}$ $$h_K(u):=\sup_{k\in K} \sum_{i=1}^d k_iu_i$$ with $u:=(u_1,...,u_i,...,u_d)\in \mathbb{R}^d$.

Under which sufficient conditions on $K$ $\sup_{k\in K} \sum_{i=1}^d k_iu_i=\max_{k\in K} \sum_{i=1}^d k_iu_i$?

• Do you think there is a simple answer on this question? – gerw Apr 22 '16 at 17:15
• I have no idea. – STF Apr 22 '16 at 17:33
• Just one set of sufficient conditions would be enough. – STF Apr 22 '16 at 18:11
• One sufficient condition is that $K$ is compact. – gerw Apr 22 '16 at 18:49