[PPL-devel] thanks

[P M Hill]

Hi Angela,

> I think that it need to add the notion of duality in the  initial page of the
> documentation, but I only find this definition for polytope (Wilde - A
> library for doing polyhedra operations - publication interne 785 - december
> 1993 - page 12).
> Can we use the same definition for general polyhedra, too?

I have been looking at the definition in Wilde for dual.
Comparing this with the definition of a polar, makes me think that what we
want is to talk about a polar (also defined by Wilde) rather than a dual.
When the polyhedra is a polytope that includes the origin, then it is
likely that a polar and a dual (wrt LP) are the same thing. I'm not
certain.
However, if we want to generalise to any polyhedra, the polar seems what
we want.
It is also clearly defined and illustrated in the text books.

Looking in Schrijver (do you have a copy of this book?) chapter 9, page
112/3, if P* is a polar of a polyhedron P, then the face-lattice of P* is
the reverse of the face-lattice of P.

Also, If C \sseq R^n is a polyhedral cone, then the polar C* is also a
polyhedra cone in R^n such that there is a 1-1 correspondence between the
faces of C and C*
In this case, a face of dimension k  in C is mapped to a face in C* of
dimension n-k.
This means that there is a correspondence between the extreme rays of P
with the facets of P* and vice-versa.

In Nemhauser, page 99, there is a theorem 5.2 that says that
if the dimension of P = {x | Ax =< b} is n and rank of A is n 
and pi \in R^n not= 0,
then
(pi,pi_0) is an extreme ray of P* iff (pi,pi_0) defines a facet of P.

Also, that if P is a full-dimensional polytope, then after normalising, P*
is also a full-dimensional polytope.
There is a nice illustration on page 102 of two such polytopes.

> [...]
> > Does anyone feels up to rewriting a specialised version of this definition
> > just for its application to rational polyhedra?
> >
> 
> ??? I can't answer this question for two reasons:
> - first I have not clear the meaning of "feels up" (sorry!)
> - moreover I don't understand what does it means that a ray is stable (even
> it seems to be
>     very easy! ;-))
> 

Sorry, I should be more careful in using colloquial expressions...
"feels up to" means (approximately) "is able to"
but it also implies that the task is hard and you may not want to...
I am sure it is hard - but maybe one of you young mathematicians can show
us an easy way to understand this - just in the context of polyhedra.

ciao,
  Pat