forum.polymake.org

Posted: **11 Dec 2015, 19:47**

I often find myself working with lattice polytopes whose vertices live in an affine hyperplane H of R^d defined by H={ x in R^d : <v,x>=1 } where v is some fixed integral vector. What is the best way to input these polytopes into polymake?

For example, suppose that the vertices of my polytope are the columns of

Code: Select all

A=
-4 -2 -5 -3
 3  2  3  2
 2  1  3  2

The vertices of this polytope live in the hyperplane defined by the integral vector v=<1,1,1>. To input my data into polymake I need to translate this hyperplane defined by e=<1,0,0>.

Banging my head against the wall, I find that A = R*B where

Code: Select all

R=              B=
-4  2 -1        1 1 1 1
 3 -1  0        0 1 0 1
 2 -1  1        0 0 1 1

Since det R = -1 I find that my original polytope is the square and I can use the columns of B to define my polytope in polymake.

Can polymake automate this process?

Posted: **16 Dec 2015, 11:17**

Hi,

one way to do this is to add an additional homogenization first:

Code: Select all

$p = new Polytope(POINTS=>ones_vector() | transpose($A));

Many properties also work fine for non-full-dimensional polytopes. But you can also project the polytope down again while preserving the lattice (as long as it is bounded and integral):

Code: Select all

> $q = ambient_lattice_normalization($p);
> print $q->VERTICES;
1 0 0
1 -1 0
1 0 1
1 -1 1

Alternatively you could also use smith_normal_form(M) directly.

Benjamin

forum.polymake.org

Can polymake help me translate my polytopes?

Can polymake help me translate my polytopes?

Re: Can polymake help me translate my polytopes?