## Motivation:

This is related to a different question I asked in April. It occurred to me while thinking about the sums of uniform random variables and it stuck in my mind because it's the special case of a more general problem:

Given the hyperplane cut of $[-1,1]^N$,$H_{N}=\{\vec{x} \in [-1,1]^N:\sum_{i=1}^N x_i = 0\}$, what is the vertex set of $H_N$?

which reduces to the original question when $N \in 2\mathbb{N}$.

## My conjecture for this problem:

I conjecture that:

\begin{equation} V_{2N}=\{\vec{x} \in \{-1,+1\}^{2N}:\sum_{i=1}^{2N} x_i = 0\} \tag{1} \end{equation}

is the smallest set such that $\mathrm{conv}(V_{2N})=H_{2N}$ where we note that:

\begin{equation} \lvert V_{2N} \rvert= {2N \choose N} \tag{2} \end{equation}

and it's relatively easy to show that:

\begin{equation} \mathrm{conv}(V_{2N}) \subset H_{2N} \tag{3} \end{equation}

So far I haven't managed to show that $\mathrm{conv}(V_{2N}) = H_{2N}$ but I have managed to show that if we define:

\begin{equation} \forall k < N, V_{2k}=\{\vec{x} \in H_{2N}:\sum_{i \in \Gamma} x_i = \sum_{j \notin \Gamma} \lvert x_j \rvert = 0 \land \{x_i\}_{i \in \Gamma} \in \{-1,+1\}^{2k} \} \tag{4} \end{equation}

then we have:

\begin{equation} \forall k < N-1, \mathrm{conv}(V_{2k}) \subset \mathrm{conv}(V_{2k+2}) \tag{5} \end{equation}

I have spent some time thinking about this problem and at this point I'm not sure how to proceed. The challenge is basically to show that $V_{2N}$ is the vertex set of $H_{2N}$.