The following definition is taken from the paper here:
Let $L$ be an $n$-dimensional lattice. The Voronoi cell $V(u)$ is then defined as the set $\{x\in \mathbb R^n:|x|\le |x-v|, \mbox{ for all }v\in L\}$. A vector $v$ is called a Voronoi vector if the hyperplane $\{x:\in \mathbb R^n:x^Tv=|v|^2/2\}$ has a nonempty intersection with $V(0)$. A Voronoi vector is called relevant if this intersection is an $n-1$-dimensional face of $V(0)$.
An alternate definition taken from the paper here:
A relevant Voronoi vector $v\in L$ is one for which $v^Tx<|x|^2$ for all $x\in L\setminus \{0,v\}$.
I want to check that these two definitions are equivalent. But I don't understand how to interpret the $n-1$ dimensional face of $V(0)$ part. Can someone prove that they are indeed equivalent?
I also wanted to know the answer to this and pieced this together. Please let me know if anything seems unclear or wrong -- it's very possible that some errors crept in.
A geometric interpretation of the condition:
First, we are going to reinterpret one of the definitions geometrically.
Note that $v^T x < xx^T$ is is equivalent via simple algebra to $||x - v/2|| > ||v/2||$:
$(x - v/2, x - v/2) > (v/2, v/2)$ $\iff$ $(x,x) - (x,v) + ||v/2||^2 > ||v/2||^2$ $\iff$ $(x,x) > (x,v)$.
Thus the statement: $v^T x < xx^T$ for all $x \in L \setminus \{ 0, v\}$ is equivalent to the statement that there are no lattice points in the closed ball $B(v/2, ||v/2||)$ other than $v$ and $0$.
I learned this from A characterization of root lattices by Dayan S.Rajana and Anil M. Shende.
Proof of equivalence of the definitions:
Let's fix a definition for Voronoi relevant:
Definition: A Voronoi relevant vector as one such that the hyperplane $H_v = \{ x : 2 (x,v) = (v,v) \}$ intersects the voronoi cell $V$ in an $n-1$ dimensional face.
We define the half spaces $H^+_s = \{ x : (x,s) \leq \frac{1}{2} (s,s) \}$, and note that $V = \bigcap_{v \in L} H^+_v$. An equivalent definition is that a $v \in L$ is voronoi relevant iff it is part of the minimal set $S \subseteq L$ such that $\bigcap_{v \in S} H^+_v = V$. (This is a fact about full dimensional polytopes defined by the intersection of half spaces.)
Here is a theorem, and useful third characterization, that we will use to connect the two definitions:
Theorem 1 (Theorem 10 in Conway/Sloane, page 477, due to Voronoi as cited in Conway/Sloane): A vector $v$ is Voronoi relevant iff $v, -v$ are the only shortest vectors in $v + 2L$.
Proof:
We will follow Conway/Sloane, but I'll add some extra explanation since they are terse.
$\Rightarrow$ Suppose that there is a $w \in v + 2 L$ with $v \not = \pm w$ and $||w|| \leq ||v||$.
We define $ t = \frac{v + w}{2}$ and $u = \frac{ v - w}{2}$. By construction, $t, u \in L$.
We are going to show that $H^+_t \cap H^+_u \subseteq H^+_v$. This will imply that $H^+_v$ is not in the minimal set of hyperplanes needed to define $V(0)$, so that $v$ is not relevant by the equivalence discussed after the definition.
Since it confused me, I want to note that $H^+_t \cap H^+_u \subseteq H^+_{t + u} = H^+_v$ is not a general fact (take $w = (100,0)$ and $v = (0,2)$, then $v \in H^+_t \cap H^+u$) -- we are going to use the assumption that $||w|| \leq ||v||$.
Consider any $x \in H^+_t \cap H^+_u$. By definition $(x,t) \leq \frac{1}{2} (t,t)$ and $(x,u) \leq \frac{1}{2} (u,u)$. By calculating $$(x,v) = (x, t + u) = (x,t) + (x,u) \leq \frac{1}{2} ( (t,t) + (u,u) )= \frac{1}{2} ((\frac{v + w}{2},\frac{v + w}{2}) + (\frac{ v - w}{2},\frac{ v - w}{2})) = \frac{1}{2} ( 2||v/2||^2 + 2||w/2||^2) = \frac{1}{2} (\frac{||v||^2 + ||w||^2}{2}) \leq \frac{1}{2} ||v||^2,$$ we learn that $x \in H_v^+$ as well.
$\Leftarrow$ If $v$ is not relevant, then we claim there is a $w \in L \setminus \{0,v\}$ such that $\frac{1}{2} v \in H_w$ or $\frac{1}{2} v \not \in H_w^+$. To see this clearly, we break things down into three cases:
In both cases, we have $\frac{1}{2} (v,w) \geq \frac{1}{2} (w,w)$, which we rewrite as $(v,w) \geq ||w||$.
This implies that $||v - 2w|| \leq ||v||$ : $||v - 2w|| = (v - 2w, v - 2w) = ||v|| - 4(v,w) + 4 ||w|| \leq ||v||$.
Moreover, as $w \not = 0, w \not = v$ we have $v - 2w \not = \pm v$. By construction $v - 2w \in v + 2 L$. Thus, $v - 2w$ contradicts the assumption on $\pm v$ being the only shortest vectors.
QED
Now we will connect your two definitions, using the geometric reformulation mentioned above:
Theorem 2: A vector $v$ is Voronoi relevant iff there are no lattice points in the closed ball $B(v/2, ||v/2||)$ other than $v$ and $0$.
Proof:
For both directions we will use Theorem 1.
$\Rightarrow$ Suppose that $v$ is Voronoi relevant. Suppose that $z \in B(v/2, ||v||/2)$ is a lattice vector, with $z = v/2 + x$, where $||x|| \leq ||v||/2$. Then $2x = -v + 2z \in v + 2L$, and $||2x|| \leq ||v||$. Thus, by theorem 1, $2x = \pm v$. If $x = v/2$ then $z = v$ and if $x = -v/2$ then $z = 0$.
$\Leftarrow$ Suppose that $w$ was a shortest vector in $v + 2L = -v + 2L$. In particular, $||w|| \leq ||v||$. Since we have that $v/2 - w/2 \in L$, by $||w|| \leq ||v||$ we have that $v/2 - w/2$ is a lattice vector inside of $B(v/2, ||v/2||)$. Thus either $v/2 - w/2 = 0$ or $v/2 - w/2 = v$; in the former case we have $v = w$ and in the latter case we have $w = -v$. Thus, by the condition of theorem 1 we have that $v$ is Voronoi relevant.
QED
I wanted to add this corollary of Theorem 1, even though it is not part of your question. The point is that each nonzero coset $L / 2L$ contributes at most $2$ relevant vectors. Since there are $2^n - 1$ such nonzero cosets, there are at most $2 (2^n - 1)$ relevant vectors. In the 'generic case,' the only vectors $x,y \in v + 2L$ with the same norm are conjugate pairs $x = -y$, so in this case each nonzero coset contributes $2$ relevant vectors. Thus, generically, the Voronoi cell has $2(2^n - 1)$ facets. This discussion is from this paper: https://www.jstor.org/stable/52019