1
$\begingroup$

1) How to find the intersecting hypervolume of 2 $k$-simplexes, each represented using a set of $(k+1)$ vertices?

2) Is it easy to check whether a given point exist in the intersecting hyperspace?

Is there any software/API that can calculate the intersecting hypervolume of 2 $k$-simplexes using only vertices? What about finding if a vertex exists in the intersecting hyperspace?
If the task is not easy. What are the challenges?

  • 2
    Sage should be able to, although I've never used it for intersections. You can view the polytope tutorial [here](http://doc.sagemath.org/html/en/thematic_tutorials/polytutorial.html), and it has an [online interface](https://cloud.sagemath.com) so you wouldn't need to download anything to give it a shot. I'll see how it works eventually and post something, if nobody beats me to it. – 2017-02-08
  • 0
    What if I give Sage so many points that they cannot be uniquely connected? In addition, does Sage support hypervolume calculation? – 2017-02-08
  • 0
    Sage can compute the convex hull of a set of points, and build (however it represents) a polytope out of it. Once you have a (Sage) polytope, you have access to its convex hull and bounding half-spaces. I would assume the intersection of two simplices is the set of points satisfying *both* sets of half-space inequalities, and again this is something Sage should be able to do, giving you the intersection polytope, and testing whether any additional points satisfy these inequalities (i.e, belong to the polytope). I'm not sure about volume, I didn't see anything with a quick look. – 2017-02-08
  • 0
    Although, evidently (see [here](http://mathoverflow.net/questions/979/algorithm-for-finding-the-volume-of-a-convex-polytope)) computing the volume of a generic convex polytope is a hard problem; I wouldn't be surprised to learn that Sage does not have this functionality. – 2017-02-08

0 Answers 0