Let $X$ be a vector space and $K \subseteq X$ be a pointed convex cone. Let $L$ denote the set of extreme rays of $K.$ The questions are: under which condition can I guarantee that $$K= cone(conv(L))?$$ Here, $cone(A)=\{\lambda x: x\in A, \; \lambda \geq 0\}$ and $conv(A)$ is the convex hull of $A.$ Any reference that treats this problem? I am particularly interested in the infinite dimensional case. Thanks in advance
[Math] Convex cone generated by extreme rays
convex-analysisconvex-conedual-conefunctional-analysisreference-request
Related Solutions
I don't know a specific reference for thid sort of question, so let me just address your two example questions. (Everything I say is discussed in the books of Kollár--Mori, Lazarsfeld, Debarre, etc.)
Let $X$ be the blowup of $\mathbf P^2$ in 9 very general points. Then the Mori cone of $X$ is well-known: its extremal rays are
the class $-K$ of the anticanonical divisor (i.e. the proper transform of the unique cubic through the 9 points)
the 9 exceptional divisors $E_1,\ldots,E_9$
the images of the $E_i$ under the action of the Cremona group: it doesn't matter what these look like, only that there are infintely many of them.
We can write any class on $X$ in the form $$ aH + \sum_{i=1}^9 b_i E_i $$ for some $a, \, b_i \in \mathbf Z$, and intersections with the exceptionals are given by $( aH + \sum_{i=1}^9 b_i E_i) \cdot E_j = -b_j$. (For classes of the third kind above other than the $E_i$ themselves, the coefficients $b_i$ are therefore nonpositive.
Since there are infinitely many classes of the third kind above, the corresponding coefficients $(b_1,\ldots,b_9)$ cannot be bounded, and so the answer to
Are the distinct intersections bound, for example to be less than or equal to 1?
is no.
On the other hand if we blow up 8 or fewer points in $\mathbf P^2$, we get a del Pezzo surface whose Mori cone has finitely many extremal rays. That means that of the infinite set of classes above, there are only finitely many in which one of the coefficients $b_i$ equals zero. So back on $X$ a given exceptional $E_i$ has nonzero intersection with all but finitely many extremal rays. Therefore the answer to
Can an extremal ray have non-zero intersection with arbitrarily many other extremal rays?
is yes.
Finally let me mention an obvious example in which one has positive answers to your questions: if $X$ is a nonsingular toric surface, then any extremal ray of the Mori cone is spanned by a torus-invariant curve. Two such curves $C_1$ and $C_2$ intersect as follows:
- if the corresponding rays span a cone of the fan of $X$, then $C_1 \cdot C_2=1$;
- otherwise $C_1 \cdot C_2=0$.
So intersections are bounded by 1, and each ray has nonzero intersection with at most 2 other rays.
Let $C^\ominus = \{ u\in E^* \,|\, \langle u,C\rangle \leq 0\}$ and let $H = \{u\in E^* \,|\, \langle u,-y\rangle \leq 0\}$.
If the interior of $C^\ominus$ meets $H$, or if $C^\ominus$ meets the interior of $H$, then we have the sum rule $\partial (\iota_{C^\ominus}+\iota_{H})= \partial \iota_{C^\ominus} + \partial \iota_{H}$. In that case, evaluating at $0$, gives
$$ (C^\ominus\cap H)^\ominus = C+H^\ominus = C+\mathbb{R}_+y\;\;\text{is closed.}$$
Another case arises when $C$ is a polyhedral cone. All the above is true in $\mathbb{R}^n$ and in Hilbert space, and likely in Banach space. I don't know about your very general setting.
Best Answer
Nothing of this sort is true for most standard cones (e.g. the natural cone in classical function spaces). Consider the following examples.
Similarly, the statement fails for many spaces of differentiable or Lebesgue integrable functions on some domain $U \subseteq \mathbb{R}^n$. (It is however true for most sequence spaces, for instance $\ell_{\mathbb{R}}^p$ with $1 \leq p < \infty$.)
To get sufficient criteria for the statement to be true, I guess one must resort to Krein–Milman type theorems (e.g. assume that the cone has a weakly compact base). For more on this, see §3.8 and §3.12 in [Jam70]. (Warning before reading Jameson's book: in the ordered vector spaces community, cone means proper/pointed convex cone — see §1.1.) In particular:
References.
[Jam70]: Graham Jameson, Ordered Linear Spaces, Springer Lecture Notes in Mathematics 141, 1970.