Cones, Extremal Rays and the Choquet Theory

Introduction

A cone is a convex set closed under nonnegative scaling, and its one-dimensional convex subsets are its rays. A ray that cannot be the sum of two vectors outside it is an extremal ray, and the extremal rays are the primitive directions of a cone in exactly the sense in which the extreme points are the primitive points of a convex set: a base cuts the cone by a hyperplane, the extreme rays of the cone become the extreme points of the base, and the theory of one is the theory of the other. This is the structure that an ordered vector space carries its order in: the positive cone, its rays, and the order that the rays generate.

The second half of the article is the Choquet theory, the integral representation of the points of a compact convex set by measures carried by its extreme points. A compact convex set is the closed convex hull of its extreme points by Krein–Milman, but that is a statement about the closure; Choquet's theorem states that every point is the barycentre of a measure concentrated on the extreme set, and it is therefore a much finer statement, because the measure remembers the proportions in which the extreme points are mixed. The extreme points are the Choquet boundary for a metrizable compact convex set, by the Choquet–Bishop–de Leeuw theorem, and the representing measure is unique exactly when the set is a simplex, by the Choquet–Meyer theorem. These are the results that make a compact convex set an integral-representation device, and they are used by the order theory of the states and the weights later in this category.

The setting is the one of Convex Sets and the Convex Hull: a real locally convex space, with compactness used where Krein–Milman is used. The cones and the ordered vector spaces are those of Ordered Vector Spaces and the Order Unit, whose state space is the principal example; the duality of a cone with its polar, and the bipolar theorem, are Duality Theory; the measure and the integral are Measure Theory and Integration, in the Foundations of Analysis category of this Part. The finite-dimensional theory of cones, polyhedral cones and their extreme rays is Convex Analysis and is cited.

Convex Cones and their Rays

Cones, the Order and the Dual

Definition. A subset $C$ of a real vector space $E$ is a convex cone when it is convex and closed under nonnegative scaling,

$$ C + C \subseteq C, \qquad \lambda C \subseteq C \ \text{for } \lambda\geq0 . $$

It is pointed when $C\cap(-C) = \{0\}$, generating when $C - C = E$, and proper when it is pointed. A pointed convex cone makes $E$ an ordered vector space by $x\leq y \iff y - x\in C$, and conversely the positive cone of an ordered vector space is a pointed convex cone.

Definition. A ray of $C$ is the set $\mathbb{R}_{\geq0}\,x = \{tx : t\geq0\}$ of nonnegative multiples of a nonzero $x\in C$. A ray $R$ is extremal when

$$ x, y\in C,\ x + y\in R \implies x, y\in R . $$

A vector $x\in C$ is an extremal generator when its ray is extremal, and the set of extremal rays is written $\operatorname{ray}(C)$.

Proposition. For a convex cone $C$ the extremal rays are exactly the one-dimensional faces of $C$; the cone is the conical hull of its extremal rays when it is closed, polyhedral and pointed; and the sum of two vectors on distinct extremal rays lies on no extremal ray.

Proof. If $R$ is a face and one-dimensional then $x+y\in R$ with $x,y\in C$ puts $x$ and $y$ in the face $R$ by the face condition, so $R$ is extremal; conversely an extremal ray is a face because a segment with endpoints in $C$ meeting the ray has its endpoints in the ray. The generation by extremal rays is the finite-dimensional Minkowski statement of Convex Analysis; the last claim is the contrapositive of the defining property.

Bases

Definition. A base of a convex cone $C$ is a convex subset $B\subseteq C$ such that every nonzero $x\in C$ is a unique positive multiple of a point of $B$; equivalently $B = C\cap f^{-1}(1)$ for a linear functional $f$ with $f>0$ on $C\setminus\{0\}$.

Proposition (extreme points of a base are extremal rays). Let $B$ be a base of a convex cone $C$. The map $x\mapsto\mathbb{R}_{\geq0}\,x$ sends the extreme points of $B$ onto the extremal rays of $C$, and it is a bijection between them.

Proof. If $b\in B$ is extreme and $x+y\in\mathbb{R}_{\geq0}b$ with $x,y\in C$, divide by $f(x+y)=1$ and write $x = f(x)\,b_1$, $y = f(y)\,b_2$ with $b_1,b_2\in B$ and $f(x),f(y)\geq0$; then $b$ is the convex combination $f(x)b_1 + f(y)b_2$, and extremality forces $x,y$ onto the ray of $b$. Conversely if the ray of $b$ is extremal and $b = tx_1+(1-t)x_2$ with $x_i\in B$, then $b$ is the sum of $tx_1$ and $(1-t)x_2$ in the cone, so both are on the ray of $b$, and being in $B$ and on that ray they equal $b$.

Proposition (the dual cone and the base of the dual). The dual cone of $C$ is

$$ C^* = \{f\in E' : f(x)\geq0 \ \text{for every } x\in C\}, $$

a closed convex cone in the dual for the weak- topology; when $C$ is closed, proper and generating with a base, the states $S = \{f\in C^* : f(u) = 1\}$ of Ordered Vector Spaces and the Order Unit are a base of $C^*$, weak- compact when the base of $C$ is compact.

Proof. $C^*$ is the intersection of the closed half-spaces $\{f : f(x)\geq0\}$, hence a closed convex cone. Every nonzero $f\in C^*$ has $f(u)>0$ when $u$ is an order unit in the interior of $C$, so $f/f(u)$ lies in $S$ and the representation is unique; the compactness is that of the polar of the order interval, which is the base.

The Choquet Theory

Measures and Barycentres

Definition. Let $K$ be a compact convex subset of a real locally convex space $E$. A probability measure $\mu$ on $K$ has a barycentre $r(\mu)\in K$, the unique point satisfying

$$ f\bigl(r(\mu)\bigr) = \int_K f\,d\mu \qquad \text{for every } f\in E' . $$

The measure represents $x$ when $r(\mu) = x$, and the set of representing measures is a convex weak-* compact subset of the probability measures on $K$.

Proposition. The barycentre exists and lies in $K$; the assignment $\mu\mapsto r(\mu)$ is affine and weak-* to weak continuous; and the point mass $\delta_x$ represents $x$.

Proof. The functional $f\mapsto\int f\,d\mu$ is linear and continuous on $E'$ with respect to the topology of uniform convergence on $K$; it is a positive functional of norm one; the point it defines lies in $K$ because $K$ is the intersection of the half-spaces containing it and a point outside would be separated from $K$ by a functional whose integral would then violate $\lvert f\rvert\leq\sup_K\lvert f\rvert$. Affineness in $\mu$ is linearity of the integral; and $\delta_x$ gives $f(x)$.

Definition. The order on the representing measures is

$$ \mu\preceq\nu \iff \mu(f)\leq\nu(f) \ \text{for every continuous convex } f:K\to\mathbb{R}, $$

and a measure is maximal when it is maximal for $\preceq$. The Choquet boundary $\partial K$ is the set of points $x$ whose point mass $\delta_x$ is maximal, equivalently the points at which every upper semicontinuous convex function that is $\leq0$ on $\operatorname{ext} K$ is $\leq0$ at $x$.

Choquet's Theorem

Theorem (Choquet). Let $K$ be a compact convex subset of a locally convex space. Every $x\in K$ is the barycentre of a maximal probability measure: there is $\mu$ on $K$ with $r(\mu) = x$ and $\mu\preceq\nu$ for every representing measure $\nu$ of $x$.

Proof. The set $M_x$ of representing measures of $x$ is nonempty (it contains $\delta_x$) and weak- compact, and the order $\preceq$ is inductive on it, an upper bound of a chain being any weak- cluster point; Zorn's lemma gives a maximal element. The delicate point is that the supremum of a chain of measures exists as a measure, which is the Riesz representation theorem of Measure Theory and Integration applied to the limit of the integrals of convex functions, and it is the step where compactness of $K$ is consumed.

Theorem (Choquet–Bishop–de Leeuw). Every maximal measure $\mu$ on $K$ is carried by the Choquet boundary, $\mu(K\setminus\partial K) = 0$. Consequently, for a metrizable $K$, where $\partial K = \operatorname{ext} K$, every $x\in K$ is the barycentre of a measure carried by the extreme points,

$$ x = \int_{\operatorname{ext} K} e\,d\mu(e) . $$

Proof. If $x\notin\partial K$ there is an upper semicontinuous convex $f$ with $f(x)>0$ and $f\leq0$ on $\partial K$; the set where $f$ attains its positive maximum is a compact convex subset of $K$ disjoint from the boundary, and a maximal measure gives it no mass, by the maximality; iterating over the boundary and using the metrizability to run a countable exhaustion yields the result. The identification $\partial K = \operatorname{ext} K$ in the metrizable case is the Choquet–Bishop–de Leeuw theorem.

The Bauer Maximum Principle and the Simplices

Theorem (Bauer). Let $K$ be compact convex and let $f : K\to\mathbb{R}$ be upper semicontinuous and convex. Then $f$ attains its maximum at an extreme point of $K$.

Proof. The set $F$ of maximisers is a nonempty compact face of $K$, and a compact convex set has an extreme point $x$; an extreme point of the face $F$ is extreme in $K$, and it maximises $f$.

Definition. A compact convex set $K$ is a simplex when every $x\in K$ has a unique maximal representing measure, equivalently when the cone of affine continuous functions together with the constants is generated by its positive part in the order of Ordered Vector Spaces and the Order Unit.

Theorem (Choquet–Meyer). For a compact convex set $K$ the following are equivalent: $K$ is a simplex; every $x\in K$ is represented by a unique maximal measure; the barycentric map from the probability measures on $\operatorname{ext} K$ to $K$ is a bijection. A simplex is affinely isomorphic to the set of probability measures on a compact Hausdorff space only in the finite-dimensional case, where it is the convex hull of affinely independent points.

Proof. Uniqueness of the maximal measure at every point is equivalent to the cone of affine functions being lattice generated, which is the statement that the dual order is a Riesz order in the sense of The Order Projection; the equivalence of the lattice condition with the uniqueness is the theorem of Choquet and Meyer. The affine isomorphism for a finite simplex is the one of Convex Sets and the Convex Hull.

Worked Cases

The Cone of Positive Semidefinite Matrices

Let $C = H_n(\mathbb{R})_+$ be the cone of positive semidefinite matrices, a pointed generating cone in the real symmetric matrices. Its extremal rays are the rays $\mathbb{R}_{\geq0}\,vv^{\mathsf{T}}$ of the rank-one positive matrices, indexed by the lines of $\mathbb{R}^n$ up to sign; the base cut out by the trace is the set of density matrices, whose extreme points are the rank-one projections, and the identification is the bijection of the base proposition. Every positive semidefinite matrix is the sum of at most $n$ rank-one positive matrices with positive coefficients, by the spectral theorem, which is the finite-dimensional instance of the conical hull of the extremal rays.

The State Space of the Continuous Functions

Let $K = \{\mu : \mu\geq0,\ \mu(X) = 1\}$ be the probability measures on a compact Hausdorff space $X$, viewed as a weak-* compact convex subset of the dual of $C(X)$; it is a simplex, and the extreme points are the point masses $\delta_x$ by the Bauer maximum principle applied to the evaluation $f\mapsto f(x)$. The barycentric map from the probability measures on the extreme points to $K$ is the identity, and the uniqueness of the representing maximal measure is the statement that a probability measure on $X$ is determined by its integrals against the continuous functions.

The Interval

Take $K = [0,1]$, a compact convex subset of $\mathbb{R}$, whose extreme points are $0$ and $1$. Every $x\in K$ is the barycentre of the unique measure $(1-x)\delta_0 + x\delta_1$, so $K$ is a simplex with one-dimensional maximal measures; this is the smallest instance of the theory and the model for the integral representation of a state of the order-unit space of Ordered Vector Spaces and the Order Unit.

Summary

A convex cone is a convex set closed under nonnegative scaling; it is pointed, generating and proper in the usual senses, and it is exactly what an ordered vector space orders by. Its rays are its one-dimensional subsets, and the extremal rays are its one-dimensional faces, exactly those rays $R$ with $x+y\in R$, $x,y\in C$, forcing $x,y\in R$. A base cuts the cone by a hyperplane, the extreme points of the base correspond bijectively to the extremal rays of the cone, and the dual cone carries the weak- compact state space as a base. On a compact convex set $K$ every probability measure has a barycentre $r(\mu)$ with $f(r(\mu)) = \int f\,d\mu$, and Choquet's theorem states that every point is the barycentre of a maximal measure; Choquet–Bishop–de Leeuw states that a maximal measure is carried by the Choquet boundary, which is the set of extreme points when $K$ is metrizable; and the Choquet–Meyer theorem states that the representing maximal measure is unique at every point exactly when $K$ is a simplex. Bauer's maximum principle states that an upper semicontinuous convex function attains its maximum at an extreme point. The cones and the order are Ordered Vector Spaces and the Order Unit, the extreme points and Krein–Milman are Convex Sets and the Convex Hull, the duality of cones is Duality Theory, the integral and the Riesz representation theorem are Measure Theory and Integration, and the polyhedral cones are Convex Analysis*.

Summary of Notation

Symbol	Meaning
$C$, $E_+$	Convex cone; positive cone of an ordered vector space
$\mathbb{R}_{\geq0}\,x$	The ray through $x$
Extremal ray	A one-dimensional face of the cone
$B = C\cap f^{-1}(1)$	A base of the cone, $f>0$ on $C\setminus\{0\}$
$C^* = \{f : f\geq0 \text{ on } C\}$	Dual cone
$r(\mu)$	Barycentre of the probability measure $\mu$
$\mu\preceq\nu$	Order of measures, $\mu(f)\leq\nu(f)$ for convex continuous $f$
$\partial K$	Choquet boundary
Simplex	A compact convex set with a unique maximal representing measure at each point