Questions tagged [extreme-points]

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Maximizing quadratic forms

Consider the maximization problem $$\text{maximize} \quad Q(x)= \sum_{i<j} \Big(\sum_{k} a_{ik}a_{jk}\Big) x_i x_j \quad \text{subject to} \quad \sum_{i}x_i^2=1,$$ and let $M$ be maximum value ...
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1answer
63 views

Extreme points of an intersection of convex set with countably many linear spaces

Let $V$ be some `nice' vector space and let $T: V\to \mathbb{R}$ be a linear functional over $V$. Define \begin{align} M= K \cap \bigcup_{i \in \mathbb{N} } \{ v \in V: T(v)=c_i \} \end{align} ...
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229 views

A Banach space where the closed unit ball is the convex hull of its extreme points

Let $X$ be a Banach space where the closed unit ball equals the convex hull of its extreme points. Is it true that this implies $X$ is reflexive?
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68 views

Extreme points of set of measures with given barycenter

Let $X$ be a convex compact metrizable subspace of a locally convex Hausdorff topological vector space, $x_0\in X$, and $P$ be the space of all Borel probability measures on $X$ with barycenter $x_0$. ...
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0answers
162 views

Does Every Extreme Point Maximize Some Linear Functional

Let $L^2$ be the set of all square-integrable functions $f:[0,1] \to [0,1]$ and $S \subset L^2$ be a closed and convex subset of $L^2$ containing the function that is constant and equal to zero. Are ...
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1answer
100 views

Regarding extreme point in a Banach space

Let $X$ be a Banach space. And let $X^* $ be the dual space of $X$. Let $E_X$ and $E_{X^*}$ denote the extreme points of the unit ball of $X$ and $X^*$. Let $x\in X$ and $|f(x)|=1$ for every $f\in E_{...
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2answers
654 views

If the closed unit ball of Banach space has at least one extreme point, must the Banach space the be a dual space?

Let $X$ be a Banach space. By Banach-Alaoglu and Krein-Milman Theorems, one can show that if $X$ is a dual space, then $X$ must have at least one extreme point of the closed unit ball. I am ...