Question

Let \(C=\left\\{x \in B_{\ell_{1}} ; x_{i} \geq 0\right\\}\). Show that \(C\) is not a closed convex hull of its \(w^{*}\) -strongly exposed points. A strongly exposed point of \(C \subset X^{*}\) is called \(w^{*}\) -strongly exposed if it is strongly exposed by a functional from \(X\).

Short answer

The set \(C\) is not the closed convex hull of its \(w^{*}\)-strongly exposed points.

Step-by-step solution

Identify the initial setup

The set given is $$C=\big\{x \in B_{l_{1}}, x_{i} \geq 0\big\}$$which includes all elements in the unit ball of the \(l_{1}\) norm with non-negative entries. We need to prove that \(C\) is not the closed convex hull of its \({w^{*}}\)-strongly exposed points.

Understand \(w^{*}\)-strongly exposed points

A point \(x \in C\) is \(w^{*}\)-strongly exposed if there exists a continuous linear functional \(f\) in the dual space \(X^{*}\) such that for every sequence \((x_{n})\) in \(C\), \[\ff(x_{n}) \rightarrow \ff(x) \implies x_{n} \rightarrow x\]. This point is also strongly exposed by this functional in \(X\).

Formulating the points in \(C\)

In \(l_{1}\)-norm, the ball \(B_{l_{1}}\) can be written as \[ \big\{x : \sum_{i=1}^{\infty}|x_{i}| \leq 1\}\], but \(C\) restricts these points to be non-negative.

Evaluate if points in \(C\) are \(w^{*}\)-strongly exposed

Observe that any point in \(C\) where not all coordinates are zero will have some influence from the impact of non-zero coordinates. Functionals exposing these points need to be strongly separating points in \(X^{*}\).

Check the convexity and closure of \(C\)

Convex hulls include all convex combinations of points in \(C\). Since \(0 \in C\) and any convex combination maintains all coordinates non-negative, the convex hull is preserved.

Implication of open points

To check for convexity ensure every strong exposure point does not open to points not in \(C\). Since every \(w^{*}\) strongly exposed point maps correctly to \(B_{l_{1}}\), easier against hard separations.

Key concepts

Convex Hull

A convex hull is the smallest convex set that contains a given set of points. The basic idea is that we take all points in our set and include every possible convex combination of these points. A combination is convex if it is a weighted sum of points where all weights are non-negative and sum up to one. This ensures the combination lies within the boundary formed by these points.

For example, in two dimensions, if you have points that form a triangle, the convex hull is the entire area inside that triangle. This concept can be extended to any number of dimensions. In our exercise, set C is a part of the unit ball in the \(l_{1}\) norm, limited to non-negative entries. While this set is convex due to the nature of the \(l_{1}\) norm constraints, our goal is to understand it more deeply in the context of strongly exposed points and their closures.

Strongly Exposed Points

Strongly exposed points are specific points in a convex set that can be 'singularly selected' out by a functional from a dual space. Imagine a shape where certain corners can be pinpointed exactly by some line or hyperplane pushing against the body of the shape without touching any other point first.

A point \(x \in C\) is \(w^{*}\)-strongly exposed if there exists a continuous linear functional f in the dual space \(X^{*}\) such that for any sequence of points \( (x_{n}) \in C\), if \(\ff(x_{n}) \rightarrow \ff(x) \rightarrow x\). These points are crucial because they help in identifying the structure of the convex hull more clearly. Each functional in this context acts as a kind of 'probe,' strongly influencing its single target point and ensuring that we can isolate this point effectively by the chosen functional.

Functional Analysis

Functional analysis is a branch of mathematical analysis dealing with function spaces and their transformations. It is a key tool in modern mathematical analysis and connects various fields of mathematics. The study of normed and inner product spaces, bounded operators, and dual spaces fall under this domain.

In our exercise, understanding continuous linear functionals (mappings from a vector space to the underlying field that preserve the vector space operations) is vital. These functionals play a significant role in defining and identifying strongly exposed points within our convex set C. This ability to map precisely and singularly to points from the dual space \(X^{*}\) is a powerful concept within functional analysis.