I'm starting to read the literature of multidimensional screening models for monopolists selling $n$ goods to a continuum of buyers with $m=n$ dimensional types, and Rochet (1987) proves that a mechanism is implementable if and only if the surplus function is convex. Let $s(\theta)$ be the surplus function defined as $\max\{ u(x,\theta) -t(\theta)\}$ where $x \in \mathbb{R}^n$, and $t$ is the tariff function the monopolist announces by the Taxation Principle.
For example, the monopolist solves the problem of maximizing the following functional (assuming 0 cost of producing the good) $$\max J(t) = \int_D t(\theta) \mathrm{d} \theta = \int_D u(x,\theta)-s(\theta) \mathrm{d} \theta $$ where $\theta \in [0,1]^n$ and $t: [0,1]^n \to \mathbb{R}$.
My question is: what is the intuition behind the fact that $s(\theta)$ must be convex? What happens if the function is piecewise linear (and hence trivially convex) but globally concave, i.e. there are kinks along null-measure sets? Can you point me to any resources?