for $\operatorname{Conv}(\typespace)$. This example is much less involved than our main result, because revenue optimal mechanisms are meaningful only when they do not overcharge any reported type and guarantee non-negative utility. This constraint can be much more stringent when imposed for all types in $\operatorname{Conv}(\typespace)$ than for $\typespace$ only.

Related Works {#sec:related}
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For multi-dimensional preferences, an allocation rule for any fixed agent is implementable if and only if it satisfies the so-called *cyclic monotonicity* property [@Rochet87]. When the type space is convex, it turns out the weaker condition of *weak monotonicity* suffices for implementability [@SY05; @AK14]. It is notable that the two solution concepts we compare in Section \[sec:mapping\] precisely correspond to the case where the type space is convex and that where it is not. However, nowhere in our arguments do we make use of this beautiful fact.

Another closely related body of work, is the literature on automated mechanism design. In automated mechanism design, mechanisms are optimized for the setting and objective automatically using information about agents’ type distributions. When this work was introduced by @conitzer2002complexity the input for this problem was an explicit description of the agents distributions, however recent work has moved towards replacing this 