whenever the feasible allocation space is downward closed, any DSIC allocation rule on a type space can be extended to its convex hull by another DSIC allocation rule.

\[thm:downward\] If the set of feasible allocations ${\mathscr{F}}$ is downward closed, for any DSIC allocation rule $\allocs$ on a type space $\typespaces$, there is a DSIC extension $\extallocs$ of $\allocs$ on $\operatorname{Conv}(\typespaces)$. If $\allocs$ is implemented with a payment rule $\pays$, $\extallocs$ can be implemented by an extension $\extpays$ of $\pays$. If $\pays$ is individually rational on $\typespaces$, so is $\extpays$ on $\operatorname{Conv}(\typespaces)$.

If we do not require the statement about individual rationality, extensibility is guaranteed by an even weaker condition, which we call *single swap feasible*.

A feasible allocation set ${\mathscr{F}}$ is *single swap feasible* (SSF) if for every agent $i$ there exists an allocation ${\boldsymbol{x}^{\mathrm{ssf}}}(i) \in {\mathscr{F}}$ such that for any $\allocs' \in {\mathscr{F}}$, $(\alloci',{\boldsymbol{x}^{\mathrm{ssf}}}_{-i}(i)) \in {\mathscr{F}}$.

Intuitively, ${\boldsymbol{x}^{\mathrm{ssf}}}(i)$ is a feasible allocation vector such that if we replace the $i^{\text{th}}$ element of this vector with the $i^{\text{th}}$ element from any other feasible allocation the resulting allocation is still feasible. If ${\mathscr{F}}$ is a product space or is downward closed, it must be SSF. [^1]

\[thm:swap\] If the set of 