Incentive compatibilty conditions for multi-item auctions
Yes, it is essentially the same idea as with just one unit.
For example, see the text leading to Proposition 14.1 in the book &quot;Auction Theory&quot; by Vijay Krishna.
Let $x$ be the vector of willingness-to-pay.
Define $U(x) = \max_z \{ q(z)x - m(z)\}$ as the equilibrium utility of type $x$. The incentive compatibility (I am quoting Krishna now) &quot;implies for all $x$,
the probability vector $q(x)$ is a subgradient of the payoff function $U$, which
is convex, at the point $x$. In other words, the vector $q(x)$ is perpendicular to
the hyperplane that supports the function $U$ at $x$ — the graph of the function $U$
lies above the hyperplane.&quot; He goes on to show that
$$U(x) = U(0) + \int_0^1 q (t \cdot x) x d t,$$
where instead of directly integrating over the type $x$ you integrate  over $t\in (0,1)$.