What is the convex hull of this cone in $\mathbb{R}^n$?

Question

Let $n \ge 3$. Define the subset $D \subseteq \mathbb{R}^n$ as follows: $x=(x_1,\dots,x_n) \in D$ if and only if all the $x_i \le 0$ , and each strictly negative values appears an even number of times (not necessarily consecutively),

For examlpe, if $n=3$, then $(-1,-1,0), (-1,0,-1) \in D$, but $(-1,-2,0),(-1,0,0)$ are not in $D$.

The set $D$ is a cone in $\mathbb{R}^n$. What is the convex hull of $D$? Can we find a nice explicit description of it?

By the way, It would also be nice to know if this convex hull is a closed subset of $\mathbb{R}^n$.

Answer 1

Let $e_i$ denote the standard unit vector. Define $$ D':= \{ x: \ x = -t(e_i+e_j), \ t\ge0, \ i\ne j\}, $$ i.e., the conical hull of vectors of the type $-(e_i+e_j)$, $i\ne j$.

Clearly, $D' \subset D$. In addition, $D \subset \text{conv} D'$. Then $\text{conv}(D) = \text{conv}D'$.

Since $0\in D'$, $\text{conv}D'$ is equal to the set of vectors of form $$ \sum_{i\ne j} -t_{ij} (e_i+e_j) $$ with $t_{ij}\ge0$. Hence $$ \text{conv}D' = \{ A \cdot t: \ t\ge0\}, $$ where $A$ is the matrix generated by all the vectors the type $-(e_i+e_j)$, $i\ne j$. This set is closed.