  We answer four questions from a recent paper of Rao and Shinkar on Lipschitz
bijections between functions from $\{0,1\}^n$ to $\{0,1\}$. (1) We show that
there is no $O(1)$-bi-Lipschitz bijection from $\mathrm{Dictator}$ to
$\mathrm{XOR}$ such that each output bit depends on $O(1)$ input bits. (2) We
give a construction for a mapping from $\mathrm{XOR}$ to $\mathrm{Majority}$
which has average stretch $O(\sqrt{n})$, matching a previously known lower
bound. (3) We give a 3-Lipschitz embedding $\phi : \{0,1\}^n \to
\{0,1\}^{2n+1}$ such that $\mathrm{XOR}(x) = \mathrm{Majority}(\phi(x))$ for
all $x \in \{0,1\}^n$. (4) We show that with high probability there is a
$O(1)$-bi-Lipschitz mapping from $\mathrm{Dictator}$ to a uniformly random
balanced function.
