P M Hill wrote:
Hi,
Thanks Le Verge's paper and other goodies arrived this morning.
[...]
I have had a look through the papers you sent although I have not read them all carefully...
Let me know if there is something I should look at and comment on today...
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?
Previously, the meaning of the word "stable" on page 9 of LeVerge was in question. I found a definition of this in a book I borrowed from the library by Christopher Witzgall "Convexity and Optimization in Finite Dimensions" BUT I haven't managed to work out what "stable" really means! The book says (page 187/8):
"We call the point X_0 \in K(f) stable if every linear (nonvertical) manifold N in R^{n+1} lying below [f] and satisfying X_0 \in \pi(N) can be separated from [f] by a nonvertical plane
E:= {(X,z) | Y^T X - z = b} in R^{n+1}:
Y^T X_1 - z_1 \leq b \leq Y^T X_2 - z_2 for all (X_1,z_1) \in [f], (X_2,z_2) \in N."
(X,z) is my ascii representation for the column vector X z Also it says that "\pi is a projection from R^{n+1} to R^n in the direction of the z-axes" "f" is a convex function and "K(f) denotes the domain of finiteness of f: K(f) := {X | f(X) < +\infty}." I'm afraid that "lying below" has another complicated definition. The definition relies on lots of other definitions (eg [f] denotes an "epigraph") and the book is not easy to read.
(The author thinks all this is easy... eg "As is easily seen every linear manifold N covering M and lying below [f] is nonvertical.")
I need help here... The stability concept was first invented by R T Rockafellar in 1963. It may be best to see how he explains it. 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! ;-))
In our devref.tex file it says:
- The dimension of the \f$\mathop{\mathrm{lin. space}}\f$ is the number of irredundant lines.
I still do not like talking about individual lines being "irredundant". I think it only has meaning in terms of an "irredundant set" Meaning that no vector can be removed without changing the system it is generating. I think that something like
- The dimension of the \f$\mathop{\mathrm{lin. space}}\f$ is the rank of any set of lines that span the space.
would be better.
Ok.
Unfortunately (or fortunately, I don't know!) my work here is almost finished. I mean that from Monday I will not work for University of Parma.
Sorry you are going! But all the best for the future.
Thanks a lot!
However I will read my mail every day and I will happy to help anyone of you!
I miss you much!
Do keep in touch and tell us of all your adventures in the outside/real world.
Of course I will do.
ciao, Pat
Ciao,
Ange.
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