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

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 $$\mathbb{R}^n$$.

• Why the downvote? Jan 13 '20 at 8:17
• What have you tried? Jan 13 '20 at 8:26
• In the example $(-1,-2,0)$ we have an even number of coordinates that are strictly negative. Then it should be in $D$, right? If $(a,b,c)$ has $a,b,c<0$, then we can write $(a,b,c)=\frac{1}{2}(2a,0,c)+\frac{1}{2}(0,2b,c)$. Since $(2a,0,c),(0,2b,c)\in D$, then $(a,b,c)$ is in the convex hull of $D$. So, the convex hull is all vectors with non-negative coordinates except the axes, where there is exactly one coordinate that is negative. But the origin is in $D$, so it is also in the convex hull. Jan 13 '20 at 8:33

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.