points in ${\mathcal{B}_{n}}$ have roughly the same “view” – even the unique point with Hamming weight $n+1$ and the boundary points which have weight $n/2+1$. More formally,

\[cor:coarse transitivity\] For all even integers $n$, and for every two vertices $x,y \in {\mathcal{B}_{n}}$ there is a bijection $f \colon {\mathcal{B}_{n}}\to {\mathcal{B}_{n}}$ such that $f(x)=y$, $f(y) = x$, and for every $z \neq w \in {\mathcal{B}_{n}}$, it holds that $$\frac{1}{20} \leq \frac{{{\sf distance}}(f(z),f(w))}{{{\sf distance}}(z,w)} \leq 20.$$

To see this, first note that ${\mathcal{B}_{n}}$ is a convex subset of ${\{0,1\}^{n+1}}$, and thus, the distances between vertices in ${\mathcal{B}_{n}}$ are the same as their distances as a subset of the cube. Now, for a given pair $x,y \in {\mathcal{B}_{n}}$, let $x' = {\psi}^{-1}(x)$ and $y' = {\psi}^{-1}(y)$, where ${\psi}$ is the function from Theorem \[thm:main\]. Define $f \colon {\mathcal{B}_{n}}\to {\mathcal{B}_{n}}$ as $f(z) = {\psi}( {\psi}^{-1}(z) \oplus x' \oplus y')$. It is easy to see that $f$ indeed satisfies the requirements of Corollary \[cor:coarse transitivity\].

#### Approximating $\boldsymbol{{\psi}}_\mathbf{i}$

We highlight another consequence of our main theorem that is perhaps somewhat surprising – the bijection ${\psi}$ of Theorem \[thm:main\] is “approximately local” in the sense that almost *all* of its output bits are essentially determined by 