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

Question

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 $K$ $\sup_{k\in K} \sum_{i=1}^d k_iu_i=\max_{k\in K} \sum_{i=1}^d k_iu_i$?