I am not an expert in convex geometry but if we define $a_i \sim \mathcal{U}([-N,N])$ where $[-N,N] \subset \mathbb{R}$ and $S_n = \sum_{i=1}^n a_i$ I suspect that for arbitrary $N \in [1, \infty) $:

- The following limit always holds true:

\begin{equation} \lim_{n \to \infty} P(\lvert S_n \rvert \leq N)=0 \tag{1} \end{equation}

- There exists a closed-form expression for the volume:

\begin{equation} \mathrm{Vol}(H_n) \tag{2} \end{equation}

I must add that I'm not merely interested in a solution but in insight. I believe this may come from a combination of analysis, combinatorics, convex geometry, and algebra. So far I am aware that $H_n$ may be understood geometrically as an $n-1$ dimensional hyperplane embedded in the hypercube $[-N,N]^n \subset \mathbb{R}^n$ and orthogonal to the vector $\vec{1}_n \in \mathbb{R}^n$ whose components are all ones. It can also be shown that $H_n$ is convex and symmetric i.e.

\begin{equation} x \in H_n \iff -x \in H_n \tag{3} \end{equation}

So far I have managed to address the discrete case by modelling it as a random walk on $\mathbb{Z}$. Basically, I managed to show that if we assume $a_i \sim \mathcal{U}([-N,N])$ where $[-N,N] \subset \mathbb{Z}$ then:

\begin{equation} \lim_{n \to \infty} P(|S_n| \leq N )=0 \tag{4} \end{equation}

\begin{equation} \lim_{n \to \infty} P(|S_n| > N )=1 \tag{5} \end{equation}

In the process I also showed that if we decompose the sum $S_n$ into its positive and negative parts:

\begin{equation} S_n = S_n^+ + S_n^{-} \tag{6} \end{equation}

then we can show that:

\begin{equation} \forall k \in [0,nN-1],P(\lvert S_n \rvert =k) > P(\lvert S_n \rvert =k+1)\tag{7} \end{equation}

In the continuous case, another question that interests me concerns the finite set:

\begin{equation} V_{2N} = \{\sum_{i=1}^{2N} x_i=0:x_i = \pm 1 \} \tag{8} \end{equation}

where $\lvert V_{2N}\rvert={2N \choose N}$.

Specifically, I'd like to know whether it can be shown that the convex hull of $V_{2N}$ satisfies:

\begin{equation} \mathrm{conv}(V_{2N}) = H_{2N} \tag{9} \end{equation}

Any references that might be useful for answering any of the above questions are welcome.

## References:

- Keith Ball. An Elementary Introduction to Modern Convex Geometry. 1997.
- Peter McMullen. Volumes of Projections of Unit Cubes. 1984.
- Ricky Liu. Laurent Polynomials, Eulerian numbers, and Bernstein’s theorem. 2013.
- J. Marengo, D. Farnsworth & L. Stefanic. A Geometric Derivation of the Irwin-Hall Distribution. 2017.

hypersimplex, scaled by a factor of $N$. When $n$ is even, its $(n-1)$-dimensional volume (normalized so that a fundamental parallelopiped in the lattice $\mathbb{Z}^n\cap H_n$ has volume 1) is $N^{n-1}A(n-1,n/2-1)$, where $A(n-1,n/2-1)$ is an Eulerian number. $\endgroup$ – Richard Stanley Apr 23 at 19:39