[PPL-devel] thanks

[P M Hill]

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...

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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?

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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.

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> 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.

> 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.

ciao,
  Pat