Can we associated to every closed pointed salient convex cone C that
is contained in a Banach space V
a hyperplane H
which intersects each ray of C exactly once?
Notes:
1. C is a cone means s C is contained in C for every positive real s
2. salient means -C intersect C = the origin
3. pointed means that C contains the origin
4. a ray of C means a ray originating at the origin and contained in C

Example: If C is the first quadrant in R^3 then we can take H to be
the set of all
x in R^3 such that the components of x sum to 1, i.e.
H = { (x,y,z) : x + y + z = 1 }.
We can think of this H as the solution to f(x) = 1
where f is the linear functional f(x) = x dot (1,1,1).

In infinite dimensions I think that we can characterize hyperplanes
as solutions of bounded linear functional equation, i.e. f(x) = some
constant.
I think that I can prove the existence of such an H in the finite
dimensional
case using the Hann Banach separation theorem. But I'm not sure,
and I'm completely lost as to how to prove this in the infinite
dimensional.

So please, if anyone has a suggestion on how to prove this (or
disprove it),
or a suggested paper to read on this, please let me know:



I am sure I am missing something obvious.

Cm wrote:
Can we associated to every closed pointed salient convex cone C that
is contained in a Banach space V
a hyperplane H
which intersects each ray of C exactly once?
Notes:
1. C is a cone means s C is contained in C for every positive real s
2. salient means -C intersect C = the origin
3. pointed means that C contains the origin
4. a ray of C means a ray originating at the origin and contained in C

Example: If C is the first quadrant in R^3 then we can take H to be
the set of all
x in R^3 such that the components of x sum to 1, i.e.
H = { (x,y,z) : x + y + z = 1 }.
We can think of this H as the solution to f(x) = 1
where f is the linear functional f(x) = x dot (1,1,1).

In infinite dimensions I think that we can characterize hyperplanes
as solutions of bounded linear functional equation, i.e. f(x) = some
constant.
I think that I can prove the existence of such an H in the finite
dimensional
case using the Hann Banach separation theorem. But I'm not sure,
and I'm completely lost as to how to prove this in the infinite
dimensional.

So please, if anyone has a suggestion on how to prove this (or
disprove it),
or a suggested paper to read on this, please let me know:



I am sure I am missing something obvious.


I don't know how you are defining a "hyperplane" in a Banach space, but
taking the characterization you give (a level set of a bounded R-linear
functional into R, or equivalently a translate of a closed subspace of
codimension 1 [where all spaces are over the reals as coefficient
field]) as the definition, existence of a hyperplane satisfying the
conditions you give is equivalent to the existence of a convex open cone
U such that C \ {0} is a subset of U and U is disjoint from -U. (If f
is the functional associated with a hyperplane as desired and f is
[w.o.l.o.g.] positive on C, then let U be the preimage under f of the
positive reals. If U is as specified, apply the Hahn-Banach separation
theorem to U and -U.) Such a U will exist if V is finite-dimensional,
as then the convex hull K of the intersection of C with the unit sphere
(not ball) of V will be compact, and so will have positive distance from
-K; U can be taken to be the open cone generated by the union of all the
open balls of radius (e.g.) 1/4 the distance from K to -K with centers
at points of K. I doubt this remains true for the infinite-dimensional
case without additional hypotheses on C, but offhand I have no
counterexample.

Hope this helps.

Robert E. Beaudoin