Let $A \subset \mathbb{R}^n$ and $M \subset \mathbb{R}^{n \times m}$ be discrete sets of cardinality $N \geq 1$.
Consider the function $f: \mathbb{R}^n \times \mathbb{R}^m \rightarrow \mathbb{R}$ defined as
$ f(x,y) := \max_{(a,B) \in A \times M} x^\top (a + B y). $
Say if $f$ is continuous in $(x,y)$, concave in the first argument $x$ and convex in the second argument $y$.
Note that it does for $N=1$ (i.e. when $A$ and $M$ just have a single element) because $f$ would be continuous and affine (i.e. both concave and convex) both in $x$ (for fixed $y$) and in $y$ (for fixed $x$).