Math Problems Vault

Problem 18.68

(i)(ii)(iii)

Find the extreme points of each of the following subsets of the plane $\mathbb {R}^{2}$ :

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(i)

$\left\{ (x, y) \mid x^{2}+y^{2} \leq 1\right\}$ ;

(ii)

$\left\{ (x, y) \mid \left|x\right|+\left|y\right| \leq 1\right\}$ ;

(iii)

$\left\{ (x, y) \mid \max \left\{ x, y\right\} \leq 1\right\}$ .

Problem 18.69

In each of the following, $B$ denotes the closed unit ball of a normed linear space $X$ .

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(i)

If $X$ contains more than one point, show that the only possible extreme points of $B$ have norm 1.

(ii)

If $X = L^{p}[a, b], 1 < p < \infty$ , show that every unit vector in $B$ is an extreme point of $B$ .

(iii)

If $X = L^{\infty }[a, b]$ , show that the extreme points of $B$ are those functions $f \in B$ such that $\left|f\right| = 1$ almost everywhere on $[a, b]$ .

(iv)

If $X = L^{1}[a, b]$ , show that $B$ fails to have any extreme points.

(v)

If $X = l^{p}, 1 \leq p \leq \infty$ , what are the extreme points of $B$ ?

(vi)

If $X = C(K)$ , where $K$ is a compact Hausdorff topological space and $X$ is normed by the maximum norm, what are the extreme points of $B$ ?

Problem 18.70

A norm on a linear space is said to be strictly convex provided that whenever $u$ and $v$ are distinct unit vectors and $0 < \lambda < 1$ , then $\left\| \lambda u+(1-\lambda ) v\right\| < 1$ . Show that the Euclidean norm on $\mathbb {R}^{n}$ and the usual norm on $L^{p}[a, b], 1 < p < \infty$ are strictly convex.

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