| T.R | Title | User | Personal
Name | Date | Lines |
|---|
| 1357.1 | | SHIRE::ALAIND | Alain Debecker @GEO DTN 821-4912 | Thu Dec 20 1990 12:55 | 37 |
| Let K be an Archimedean convex cone in R^2, by which I mean that K is a
subset of R^2 such that
(1) if x and y are in K then x+y is in K,
(2) if x is in K and t>=0 then tx is in K,
(3) if x and -x are both in K then x=0,
(4) if ty-x is in K for all t>0 then -x is in K.
Then there are vectors u and v such that K = { su + tv : s, t >= 0 }.
===========================================================================
The key is connexity: the cone being convex is also connected by arc, thus
connected.
Consider the projection p on the unit circle defined on R�-{0} by:
x
p(x) = ---------
|| x ||
This projection being continuous, the image p(K) of the cone on the unit circle
is also connected.
The only connected subsets of the circle are continuous arcs. The full circle
being excluded by (3), this means that p(K) is an arc of the form (u,v), where
u and v are the extremities of the arc.
To show that u is in K, apply (4) to y = v and x = -u. In facts the line D =
{ tv+u | t>0} is such that p(D) = ]u,v[. Thus, by (2), D is contained in K
and u must be in K by (4). The same applies to v, so that p(K) is the arc
[u,v].
Therefore any point of the form su + tv is also in K, provided that s and t
are strictly positive.
Conversely, take any x in K. Its projection p(x) is on the arc (u,v), and
then we can write p(x) = au + bv, with a and b positive. Let s = a ||x|| and
t = b ||x|| so that x = su + tv, with s and t positive.
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| 1357.2 | | GUESS::DERAMO | Dan D'Eramo | Thu Dec 20 1990 14:38 | 11 |
| Someone on usenet noted that the definition also allowed
K to be the empty set or the singleton set containing
only the origin. In the latter case K = { su + tv : s, t >= 0 }
with u = v = (0,0).
re .-1
You twice use "positive" where I think you mean
"nonnegative", otherwise that looks right.
Dan
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| 1357.3 | | SHIRE::ALAIND | Alain Debecker @GEO DTN 821-4912 | Fri Dec 21 1990 06:15 | 19 |
| > re .-2
>
> You twice use "positive" where I think you mean
> "nonnegative"
A matter of culture. In many countries, zero is taken as a
number both positive and negative, and the sentence "x is
smaller than y" means x <= y. In US and UK, zero is neither
positive nor negative, and "x is smaller than y" doesn't
allow for equality.
Nevertheless, you are right. I was lousy on the edges:
Note that the condition (4) is equivalent to the fact that
the cone is closed. An *open* set verifying condition (1)
to (3) is of the form K = { su + tv | s,t > 0 }. The same
demonstration holds: this time, the intersection of K and
the cone is open, thus of the form ]u,v[ instead of [u,v].
|
| 1357.4 | But this is English, right? | CHOVAX::YOUNG | Give peace a chance. | Fri Dec 21 1990 11:38 | 7 |
| Re .4:
But these are fairly precise terms in English. For instance,
"Positive" in English has a precise meaning, ie. "Greater than
zero."
-- Barry
|
| 1357.5 | Let he who is without sin(x) cast the first die | VMSDEV::HALLYB | The Smart Money was on Goliath | Fri Dec 21 1990 12:37 | 5 |
| Re: .4 [Re: .4 [Re: .4 [Re: .4 [Re: .4 [Re: .4 [ Re: .4 [Re: .4 [...
> But these are fairly precise terms in English. For instance,
As opposed to "unfairly precise"?
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| 1357.6 | Precisely! | GUESS::DERAMO | Dan D'Eramo | Fri Dec 21 1990 13:55 | 5 |
| More like, as in "rather precise". More precise than
your usual English term, but not as precise as something
like "compact Hausdorff space".
Dan
|
| 1357.7 | A:x [Mathematician(x) -> Life_of_Nit_Picking(x)] | CHOVAX::YOUNG | Give peace a chance. | Fri Dec 21 1990 15:46 | 12 |
| Yeah, what Dan said.
Really though, my lousy spelling and usual fumble-fingered Notes
replies aside, when you discuss mathematics, you HAVE to be precise.
Mathematics more than any other field inherently relies on precise
terminology and precise statements.
What is, in polite conversation, merely a nit that only an irritating
pest would point out, in mathematical discussions can easily become a
pivotal issue or the fatal 'flaw' in a critical proof.
-- Barry
|
| 1357.8 | | SHIRE::ALAIND | Alain Debecker @GEO DTN 821-4912 | Fri Dec 28 1990 09:03 | 7 |
| Of course "positive" has a precise meaning in English, and I confess to
make a linguistic mish-mash from time to time. You know, just like when
you ask an Englishman why he is driving on the left side of the road and
he answers "because it is the right side".
Nevertheless in this precise case the demonstration was simpler with an
open cone.
|