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Computing the volume of a polytope that is not full-dimensional

asked 2013-11-26 04:29:09 -0500

Samsa gravatar image

Let us assume we are given a polytope in $\mathbb{R}^d$ which is $k$-dimensional. How can we compute its $k$-dimensional volume in Sage?

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answered 2013-12-03 23:51:26 -0500

slelievre gravatar image

I would do the following:

  1. Translate the polytope to have one vertex at the origin. This way it lives in a $k$-dimensional vector subspace, not only affine subspace.

  2. Find an orthonormal basis of this subspace.

  3. Express the polytope in that basis.

  4. Now work in $\mathbb{R}^k$ and find the volume of the polytope.

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Asked: 2013-11-26 04:29:09 -0500

Seen: 196 times

Last updated: Dec 03 '13