What does mean $\mathrm{conv}\{\{+1, -1\}^d\}$

Question

What does mean $\mathrm{conv}\{e_1 , -e_l , \ldots , e_d , -e_d\}$ and $\mathrm{conv}\{\{+1, -1\}^d\}.$

I could not understand, need a simple explanation.

Answer 1

Let us consider the case $d=3$.

The set $\{\{+1, -1\}^d\}$ is the set of all triples (there are $2^3=8$ of them)

$$\{(-1,-1,-1), (-1,-1,1), (-1,1,-1), \cdots (1,1,1) \}$$

i.e., the vertices of a cube.

More generally, $\{+1, -1\}^d$ is the set of vertices of a hypercube in $\mathbb{R^d}$.

As a consequence:

$$\mathrm{conv}\{\{+1, -1\}^d\}$$ is the "solid hypercube" in $\mathbb{R}^d$.