## Wednesday, July 20, 2005

### separating hyperplane for an n-dimensional unitary polytope

From sci.math on 13-JUL

Why don't we just say "polyhedra" instead of saying "convex polyhedra"? Isn't every polyhedron convex by definition(an intersection of affine
halfspaces)?

A poster replied

Well, it's really a terminology issue. Some people don't consider polyhedra convex. So for example this person's site: http://www.korthalsaltes.com/ has lots of non-convex polyhedra (eg: great icosahedron).
But yeah, in discrete geometry the definition necessitates convexity.

This is the link.
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Frame the question of ZCD in terms of the polyhedron. Frame a version of Farkas lemma for the type of polyheda. What does the unitary polyhedron mean geometrically. What does its dual mean?
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Draw the vectors in the (n+d) dimensional space.
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Approach by paths on hypergraphs.
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Given a colletion of hyperplanes, all of which pass through the origin, find if there exists a second point which lies on all the planes. Applying a variant of Farkas lemma(version II in ziegler's book), this question is equivalent to finding if there exists a row vector c in m-dimensional dual space with cA >= 0 and c < 0.
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What does it mean by saying that there exists a vector c which has a positive dist with every column of the {0,-1,+1} matrix A?
get the farkas lemma for separability.