zonotope-props {zonohedra}

zonotope properties

Description

Get some important boolean properties of a zonotope.

pointed means that 0 is a vertex of the zonotope. salient means that 0 is in the boundary of the zonotope. So pointed implies salient, but not the reverse.

A zonotope has an associated convex cone - allow the coefficients of the generators to be any non-negative numbers. For convex cones, pointed means that the cone is in an open linear halfspace (except for 0). And salient means that the cone is in a closed linear halfspace (the cone may contain a line).

In terms of generators (of both zonotopes and convex cones), pointed means that the generators are in an open linear halfspace (except for 0 generators). And salient means that the generators are in a closed linear halfspace.

Usage

## S3 method for class 'zonotope'
is_pointed( x )

## S3 method for class 'zonotope'
is_salient( x )

Arguments

x

a zonotope object - a zonohedron, a zonogon, or a zonoseg

Details

For a zonohedron, if 0 is in the interior of an edge or a facet, then the zonohdron is salient but not pointed.
For a zonogon, if 0 is in the interior of an edge, then the zonogon is salient but not pointed.
For a zonoseg, both pointed and salient are equivalent to 0 being a boundary point. And this is equivalent to all the non-zero generators having the same sign (all negative or all positive).

Value

TRUE or FALSE

References

Zonohedron - Wikipedia.
https://en.wikipedia.org/wiki/Zonohedron

See Also

zonohedron(), zonogon(), zonoseg()

Examples

zono1 = zonoseg( c(1,-2,3,0,-3,-4) )

is_pointed( zono1 )
# [1] FALSE

is_salient( zono1 )
# [1] FALSE