# What does mean conv{e_1,−e_l,…,e_d,−e_d} and conv{{+1,-1}^d}. I could not understand, need a simple explanation.

Lyla Carson 2022-09-23 Answered
What does mean $\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{v}\left\{{e}_{1},-{e}_{l},\dots ,{e}_{d},-{e}_{d}\right\}$ and $\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{v}\left\{\left\{+1,-1{\right\}}^{d}\right\}.$
I could not understand, need a simple explanation.
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Cremolinoer
Let us consider the case d=3.
The set $\left\{\left\{+1,-1{\right\}}^{d}\right\}$ is the set of all triples (there are ${2}^{3}=8$ of them)
$\left\{\left(-1,-1,-1\right),\left(-1,-1,1\right),\left(-1,1,-1\right),\cdots \left(1,1,1\right)\right\}$
i.e., the vertices of a cube.
More generally, $\left\{+1,-1{\right\}}^{d}$ is the set of vertices of a hypercube in ${\mathbb{R}}^{\mathbb{d}}$
As a consequence:
$\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{v}\left\{\left\{+1,-1{\right\}}^{d}\right\}$
is the "solid hypercube" in ${\mathbb{R}}^{d}$