Understanding the geometry of $H_{n}=\{\vec{x} \in [-N,N]^n:\sum_{i=1}^n x_i = 0\}$

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 $$N \in [1, \infty)$$:

1. The following limit always holds true:

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

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

$$$$\mathrm{Vol}(H_n) \tag{2}$$$$

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.

$$$$x \in H_n \iff -x \in H_n \tag{3}$$$$

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:

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

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

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

$$$$S_n = S_n^+ + S_n^{-} \tag{6}$$$$

then we can show that:

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

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

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

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:

$$$$\mathrm{conv}(V_{2N}) = H_{2N} \tag{9}$$$$

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

References:

1. Keith Ball. An Elementary Introduction to Modern Convex Geometry. 1997.
2. Peter McMullen. Volumes of Projections of Unit Cubes. 1984.
3. Ricky Liu. Laurent Polynomials, Eulerian numbers, and Bernstein’s theorem. 2013.
4. J. Marengo, D. Farnsworth & L. Stefanic. A Geometric Derivation of the Irwin-Hall Distribution. 2017.
• Isn't $P(S_n=0)$ always zero? It's the volume of a codimension 1 set. If you are interested in $P(S_n\in[-N,N])$ you could use a continuous local limit theorem for an asymptotic, or the CDF of the Irwin-Hall distribution for an exact formula. – Dap Apr 21 at 9:24
• $H_n$ is an example of a 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. – Richard Stanley Apr 23 at 19:39
• @RichardStanley Thank you for pointing this out. I found this particular publication: 'Laurent Polynomials, Eulerian numbers, and Bernstein’s theorem' arxiv.org/pdf/1309.3354.pdf – Aidan Rocke Apr 23 at 23:03