2
$\begingroup$

Consider a finite reflection group acting on $\mathbb{R}^N$. Pick a vector $x \in \mathbb{R}^N$ and look at the convex hull of its orbit. This is a polytope.

There are some interesting particular cases. For example, the unit $l_1$ ball can be seen as the convex hull of the orbit of the vector $(1, 0, \ldots, 0)$ under the action of the group of signed permutations. Likewise, the $l_\infty$ can be seen as the convex hull of the orbit of the vector $(1, 1, \ldots, 1)$ under the same action.

Are these polytopes well studied? What can be said about them? I would appreciate some references.

$\endgroup$
3

1 Answer 1

1
$\begingroup$

You just ask for Wythoff's kaleidoscopical construction method. Even so it usually is applied to edge sizes being either zero or unity only (twice the distance of the seed point to the respective Mirror, as then all generated edges would have the same size), it well can be applied to any distance too. Even negative distances would work. - And definitely it applies to any dimension.

--- rk

$\endgroup$

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .