Question

Show that every standard unit vector \(e_{i}\) is a strongly exposed point of \(B_{\ell_{1}}\).

Short answer

The functional \( f(x) = x_i \) uniquely maximizes at \( e_i \), proving \( e_i \) is a strongly exposed point of \( B_{\ell_1} \).

Step-by-step solution

Understand the Problem

Given the standard unit vector in the form of \( e_i = (0, 0, ..., 1, ..., 0) \) where 1 is located at the i-th position, determine if it is a strongly exposed point of the \( \ell_1 \)-unit ball, \( B_{\ell_1} \).

Define Strongly Exposed Point

A point \( x \) in a convex set \( C \) is said to be a strongly exposed point if there exists a functional \( f \) such that for all sequences \( x_n \in C \) converging to \( x \), the functional \( f(x_n) \) strictly decreases as \( x_n \) moves away from \( x \).

Identify the Functional

Consider the functional \( f(x) = x_i \). This functional will attain its maximum value at \( e_i \), that is, \( f(e_i) = 1 \).

Show Uniqueness

This functional attains its maximum value of 1 only at the point \( e_i \). Suppose there is another vector \( y = (y_1, y_2, ..., y_i, ..., y_n) \) where \( f(y) = 1 \). This would imply \( y_i = 1 \) and since \( \sum |y_j| \leq 1 \), all other \( y_j =0 \). Therefore, \( y = e_i \), proving the uniqueness.

Conclusion

Since \( f(x) \), the functional defined above, uniquely maximizes at \( e_i \), \( e_i \) is a strongly exposed point of \( B_{\ell_1} \) by definition.

Key concepts

strongly exposed point

In functional analysis, a strongly exposed point is a specific type of point within a convex set. It often relates to the behavior of functionals. By definition, a point x in a convex set C is strongly exposed if there exists a supporting hyperplane or functional that uniquely attains its maximum at this point.
Imagine navigating a 3D shape; if you find just one spot where a plane touches it, that could be a strongly exposed point.
Essential characteristics include:

• Existence of a unique supporting functional.
• Uniqueness in attaining the maximal value.

It helps understand the structure and extremities of convex sets. In our exercise, each unit vector in the \( \ell_1 \) unit ball is a strongly exposed point.

functional

A functional is a map from a vector space to the field of scalars (real or complex numbers). It's a foundational concept in functional analysis.
Think of it as a special kind of function that takes a vector as input and produces a single scalar as output.

• Mathematically, if V is a vector space over the field F, a functional f maps each element of V to an element of F: f: V → F.
• In the given problem, the functional f(x) = x_i was used.

This specific functional picks out the i-th component of the input vector.

convex set

A convex set is a subset of a vector space that contains all line segments between any two points within the set. This implies no indentations or holes.

• If you pick any two points in a convex set, the line segment joining them also lies entirely within the set.
• Examples include simple shapes like line segments, circles, and convex polygons.

In our exercise, we deal with the convex set of the \( \ell_1 \) unit ball. This set is a special case and can be visualized in lower dimensions as a diamond shape.

unit vector

A unit vector is a vector with a magnitude of exactly 1. It's often used to describe directions in space because it has no scale distortion.

• In the context of the problem, standard unit vectors have exactly one component equal to 1 and all others equal to 0.
• For example, in \( \mathbb{R}^3 \), one of the unit vectors is \( (1, 0, 0) \).

These vectors are fundamental building blocks in any vector space, providing a basis for more complex constructions. The exercise uses standard unit vectors like \( e_i = (0, 0, \ldots, 1, \ldots, 0) \).

unit ball

A unit ball in a normed vector space is the set of all vectors whose norm (or length) is less than or equal to one. It provides a way to visualize the 'size' of elements within the space.

• For \( \ell_1 \)-norm, the unit ball is defined as \( \{ x \in \mathbb{R}^n : \sum_{i=1}^{n} |x_i| \leq 1 \} \).
• In 2D, this looks like a diamond shape rather than a circular ball.

This geometric object helps understand the boundaries and limits of vectors within the normed space. Our task explores the strongly exposed points of the \( \ell_1 \) unit ball.