samples (e.g. from past observations or market research) from an underlying distribution, whereas the true distribution may be supported on a continuum of types. This approach motivates the question we study in this work.

#### Questions we study.

In this work we aim to answer the question: when one has a mechanism that is DSIC on a discretized subset of a type space, can one always find a mechanism that has the same behavior on the subset and yet is DSIC on the whole type space? To make the question more concrete, we study the natural case where the whole type space is the convex hull of the discrete subset. To make the presentation easier, in the following we denote by $\typespaces$ the discrete subset of types, and $\operatorname{Conv}(\typespaces)$ its convex hull.

We consider the question a fundamental one for the second approach to mechanism design that we described above. When one optimizes for a mechanism with IC constraints imposed only on $\typespaces$, if the resulting mechanism cannot be *extended* to the original type space, it loses incentive guarantees when put to use.

One objection may be that, given a mechanism that is DSIC on $\typespaces$, one may always run 