Let $K \subset \mathbb{R}^d$ be a compact, convex set. It could be uniquely determined by its support function (for $u$ on the unit-sphere $S^{d-1}$), given by $$h_K(u) = \sup \{ \sum_{i=1}^d x_i u_i: x \in K\},$$

where $x_i$ is the $i^{th}$ component of the vector $x$ and similarly for $u$. I was wondering if there are certain convex sets for which the summation and supremum could be interchanged, in which case $h_K(u)$ would be additive in the components of $x$. (or sets for which $h_K(u)$ could be approximated with matching upper and lower bounds that are additive ?)