   Proving closedness of intersection of half-spaces
  Hi!I'm writing a small CSG library and I need to solve the following problems.How to quickly determine if intersection of closed half-spaces forms a closed  convex polyhedron?And if the intersection of those planes is a solution to the corresponding linear programming problem, how do I find if the problem has:1) no solutions;2) one solution;3) infinite number of bounded solutions;4) infinite number of unbounded solutions?Probably, that makes no sense, because I'm no expert at this.Sorry for my English.Thanks.EDIT: changed absurd 'Proving convexity of intersection of half-spaces'to 'Proving closedness of intersection of half-spaces'.   
  By "closed" I think you really mean "bounded?"("Closed" has to do with whether a set contains its boundary; it's the difference between strict and nonstrict inequalities; just Wikipedia it if you're interested.)Anyway, I'm interested in your question too because this is a subject I also have been working on learning.  I may be able to help a little:1 - I think Farkas' Lemma answers some of your questions, in particular #1.2 - Say you have the constraintsA x + b <= 0 .Now lift your problem up one dimension to consider the problem,min zs.t.z (1,1,...,1) >= A x + b .This minimizes the worst violation of any constraint.  This problem is always feasible, but may be unbounded.  Now say it's bounded and that "z*" is the minimum.  Then, if z* is negative, all your constraints are feasible; if it's positive then at least one isn't.  If it's unbounded then what you want to know is whether it's unbounded from above or below.3 - As for your question #3:  This happens when your cost vector is in the orthogonal complement to the subspace spanned by the active constraint normals.  If you're using the simplex algorithm, I think this is the null space of the currently-active-constraint matrix...[EDIT: Oh yeah: IIRC there are efficient algorithms for finding intersections of halfplanes in the special cases of 2d and 3d, so if you're only concerned with low dimensional spaces you may be able to do things more efficiently.]   ;  Yes, i meant 'bounded' polyhedrons (which have finite, limited volumes).Thanks!