Extreme points of convex sets

If $(X, \Omega , \mu )$ is a $\sigma$-finite measure space and $1 < p < \infty$, then the set of extreme points of $\ball L^p(\mu )$ is $\left\{ f \in L^p(\mu ) : \left\| f\right\|_{p} = 1 \right\}$.

If $(X, \Omega , \mu )$ is a $\sigma$-finite measure space, the set of extreme points of $\ball L^1(\mu )$ is $\left\{ \alpha \chi_E : E \text{ is an atom of } \mu , \ \alpha \in \mathbb {F}, \text{ and } \left|\alpha \right| = \mu (E)^{-1} \right\}$.

If $(X, \Omega , \mu )$ is a $\sigma$-finite measure space, the set of extreme points of $\ball L^\infty (\mu )$ is $\left\{ f \in L^\infty (\mu ) : \left|f(x)\right| = 1 \text{ a.e. } [\mu ] \right\}$.

If $X$ is completely regular, the set of extreme points of $\ball C_b(X)$ is $\left\{ f \in C_b(X) : \left|f(x)\right| = 1 \text{ for all } x \right\}$. So $\ball C_{\mathbb {R}}[0,1]$ has only two extreme points.

Let $X$ be a totally disconnected compact space. (That is, $X$ is compact and if $x \in X$ and $U$ is a neighborhood of $x$, then there is a subset $V$ of $X$ that is both open and closed and such that $x \in V \subseteq U$. The Cantor set is an example of such a space.) Show that $\ball C(X)$ is the norm closure of the convex hull of its extreme points.

Show that $\ball l^1$ is the norm closure of the convex hull of its extreme points.

Show that if $X$ is locally compact but not compact, then $\ball C_0(X)$ has no extreme points.

Let $K$ be convex and let $T : K \to \mathcal{Y}$ be an affine map. If $y$ is an extreme point of $T(K)$ and $x$ is an extreme point of $T^{-1}(y)$, then $x$ is an extreme point of $K$.

If $\mathcal{H}$ is a Hilbert space and either $T$ or $T^*$ is an isometry, show that $T$ is an extreme point of the closed unit ball of $\mathcal{B}(\mathcal{H})$.

If $X$ is compact, show that for each $x$ in $X$, $\delta_x$ is an extreme point of $P(X)$ and $\alpha \delta_x$, $\left|\alpha \right| = 1$, is an extreme point of $\ball M(X)$.

If $X$ is a compact connected topological space that is not a singleton, show that $C(X)$ is not the dual of a Banach space. (For $C_{\mathbb {R}}(X)$ this is a consequence of Exercise 7.4. For the complex case, show that if $C(X)$ is a dual space, then $C_{\mathbb {R}}(X)$ is a weak* closed real linear subspace of $C(X)$.)

Let $\mathcal{A}$ be the closure in $C(\partial \mathbb {D})$ of the polynomials in $z$ and let $K = \left\{ \phi \in \mathcal{A}^* : \phi (1) = \left\| \phi \right\| = 1\right\}$. Show that $\text{ext}\, K = \left\{ \delta_z : \left|z\right| = 1\right\}$.

Using the terminology of the preceding exercise, show that the extreme points of ball $\mathcal{B}(\mathcal{H})$ are the maximal partial isometries. (See Exercise V.7.10.)

Show that the extreme points of $\{ A \in \mathcal{B}(\mathcal{H}) : 0 \leq A \leq 1\}$ are the projections.

Let $K$ be a convex set in a linear space $L$. A subset $A\subset K$ is said to be extreme if every segment lying in $K$ and with midpoint in $A$ lies entirely in $A$. Prove that the intersection of any family of extreme subsets is either empty or an extreme subset.

Let $K$ be a compact convex set. Prove that the collection of closed extreme subsets of it (see Problem 329), ordered by inclusion, has a minimal element.

Let $K$ be a closed convex bounded subset of an LCS $L$, and $A$ a minimal element of the family of closed extreme subsets of $K$ (see Problem 330). Prove that $A$ consists of a single point.

Prove that a compact convex set $K$ in an LCS $L$ has at least one extreme point.

Let $K$ be the collection of all positive (see Problem 643) operators with trace $1$ on a finite-dimensional Hilbert space $H$. Prove that $K$ is a convex compact set, and find the extreme points of $K$.

Prove that the closed unit ball of $L^{1}$ (relative to Lebesgue measure on the unit interval) has no extreme points but that every point on the "surface" of the unit ball in $L^{p}(1 < p < \infty )$ is an extreme point of the ball.

Determine the extreme points of the closed unit ball of $C$, the space of all continuous functions on the unit interval, with the supremum norm. (The answer depends on the choice of the scalar field).

Let $K$ be the smallest convex set in $\mathbb {R}^{3}$ that contains the points $(1,0,1),(1,0$, $-1)$, and $(\cos \theta , \sin \theta , 0)$, for $0 \leq \theta \leq 2 \pi$. Show that $K$ is compact but that the set of all extreme points of $K$ is not compact. Does such an example exist in $\mathbb {R}^{2}$?

Suppose $K$ is a compact convex set in $\mathbb {R}^{n}$. Prove that every $x \in K$ is a convex combination of at most $n+1$ extreme points of $K$. Suggestion: Use induction on $n$. Draw a line from some extreme point of $K$ through $x$ to where it leaves $K$. Use Exercise 1.

Let $\left\{ u_{1}, u_{2}, u_{3}, \ldots \right\}$ be a sequence of pairwise orthogonal unit vectors in a Hilbert space. Let $K$ consist of the vectors 0 and $n^{-1} u_{n}(n \geq 1)$. Show that

If $0 < p < 1$, every $f \in L^{p}$ (except $f = 0$) is the arithmetic mean of two functions whose distance from 0 is less than that of $f$. (See Section 1.47). Use this to construct an explicit example of a countable compact set $K$ in $L^{p}$ (with 0 as its only limit point) which has no extreme point.

Suppose that $p$ is an extreme point of some convex set $K$, and that $p = t_{1} x_{1}+\cdots +t_{n} x_{n}$, where $\sum t_{i} = 1, t_{i} > 0$ and $x_{i} \in K$ for all $i$. Prove that $x_{i} = p$ for all $i$.

Supply the details of the proof that every extreme point of $P$ has its support at a single point. (This refers to the end of proof of Theorem 5.10).

Suppose $A$ is a closed subspace of $C(S)$, where $S$ is a compact Hausdorff space; suppose $\mu$ is an extreme point of the unit ball of $A^{\perp }$; and suppose $f \in C(S)$ is a real function such that

for every $g \in A$. Prove that $f$ is then constant on the support of $\mu$. (Compare with Theorem 5.7). Show, by an example, that the conclusion is false if the word "real" is omitted from the hypotheses.