Question

Let \(A\) be a subset of a metric space \((X, \rho)\). We say that a point \(x \in A\) is diametral if \(\sup \\{\|x-y\| ; y \in A\\}=\operatorname{diam}(A)\) Define \(A=\\{x(t) \in C[0,1] ; 0=x(0) \leq x(t) \leq x(1)=1\\}\). Show that every point of \(A\) is a diametral point.

Short answer

Every function \( x(t) \in A \) has a supremum distance of 1 from some other function in A, so every point of A is diametral.

Step-by-step solution

- Understanding the set A

First, recall what the set A represents: \[ A = \{ x(t) \text{ in } C[0,1] ; 0=x(0) \text{ and } x(t) \text{ is non-decreasing from } x(0)=0 \text{ to } x(1)=1 \}. \]This means that A consists of all continuous functions on the interval [0,1] that start at 0, end at 1, and are non-decreasing.

- Understanding the Metric and Diameter

Consider the metric \( \rho \) in the space \( C[0,1] \), which is usually the supremum metric: \[ \rho(f,g) = \sup_{t \in [0,1]} |f(t) - g(t)|. \]The diameter of set A, \( \operatorname{diam}(A) \), under this metric is: \[ \operatorname{diam}(A) = \sup_{f,g \in A} \rho(f,g). \]

- Identify the Diameter

To find \( \operatorname{diam}(A) \), note that the largest possible distance under the supremum metric for functions in A is between the function that is constantly 0 and the function that is constantly 1. Thus, \[ \operatorname{diam}(A) = 1. \]

- Define Diametral Point Criteria

A function \( x(t) \in A \) is diametral if \[ \sup_{y \in A} \rho(x(t),y) = \operatorname{diam}(A). \]Given that \( \operatorname{diam}(A) \) is 1, the criterion for a function \( x(t) \in A \) to be diametral is \[ \sup_{y \in A} \rho(x(t),y) = 1. \]

- Show Every Point in A is Diametral

To show that every function \( x(t) \in A \) satisfies this criterion, note that \[ \rho(x(t),0) = \sup_{t \in [0,1]} |x(t) - 0| = |x(1)| = 1, \]since \( x(t) \) is non-decreasing and reaches 1 at \( t = 1 \). Similarly, \[ \rho(x(t),1) = \sup_{t \in [0,1]} |x(t) - 1| = 1 - 0 = 1. \]As both values are 1, each function \( x(t) \in A \) meets the criterion.

Key concepts

Metric Space

A metric space is a set where a metric (or distance function) is defined. This distance function, typically denoted as \(\rho\) or \(d\), measures the 'distance' between any two elements in the set. A metric space is written as \((X, \rho)\), where \(X\) represents the set of elements and \(\rho\) represents the metric. For example, in the Euclidean space, the metric \(\rho(x,y)\) is the standard distance between points \(x\) and \(y\). The function \(\rho\) must satisfy three key properties:

• Non-negativity: \(\rho(x, y) \geq 0\) for all \(x, y \in X\), and \(\rho(x,y) = 0\) if and only if \(x = y\).
• Symmetry: \(\rho(x, y) = \rho(y, x)\) for all \(x, y \in X\).
• Triangle inequality: \(\rho(x, y) \leq\rho(x, z) + \rho(z, y)\) for all \(x, y, z \in X\).

Metric spaces provide a formal framework for discussing and analyzing geometric concepts across various fields, including mathematics, physics, and computer science.

Supremum Metric

The supremum metric, also known as the uniform metric or infinity norm, is a way of measuring the 'distance' between functions. For two functions \(f\) and \(g\) defined on the same domain, the supremum metric \(\rho\) is given by:
\[ \rho(f,g) = \sup_{t \in \[0,1\]} |f(t) - g(t)|. \] This metric calculates the largest absolute difference between the values of \(f\) and \(g\) over the domain. The term \(\sup\) stands for 'supremum' or the least upper bound. It is particularly useful when comparing continuous functions because it focuses on the maximum deviation at any point in their domain. For example, if \(f(t) = t\) and \(g(t) = t^2\) over \([0, 1]\), then the supremum distance \(\rho(f,g)\) is the maximum distance between \(t\) and \(t^2\). This metric is commonly used in spaces like \(C[0,1]\), the set of continuous functions on the interval \([0,1]\).

Diameter of a Set

The diameter of a set \(A\) in a metric space \((X, \rho)\) measures the 'size' of the set in terms of the maximum distance between any two elements of the set. Formally, the diameter of set \(A\) is defined as:
\[ \operatorname{diam}(A) = \sup_{x, y \in A} \rho(x, y). \] This means that you take the supremum (the least upper bound) of all possible distances between pairs of elements in \(A\). For example, if \(A\) consists of the points \( \{0, 1, 2, 3\} \) in the Euclidean space with the usual metric \( \rho(x,y) = |x - y| \), the diameter would be the distance between the smallest and largest points, which is 3. In many applications, determining the diameter of a set helps to understand its 'spread' and how 'far' elements of the set can be from each other. This concept is especially useful in analysis and topology.

Diametral Point

A diametral point in a set \(A\) within a metric space \((X, \rho)\) is a special type of point that achieves the diameter of the set. Formally, a point \(x \in A\) is considered diametral if:
\[ \sup_{y \in A} \rho(x, y) = \operatorname{diam}(A). \] This means that the maximum distance from \(x\) to any other point in \(A\) is equal to the diameter of \(A\). Essentially, these points are crucial because they represent the 'extreme' values that determine the overall size or spread of the set. In the context of the exercise provided, all functions \(x(t)\) in the set \(A\) are shown to be diametral points because they each reach the maximum distance (in the supremum metric) from other points in the set. This captures a distinctive feature of the set and provides insight into its structure, demonstrating how diametral points define essential properties of geometric and functional spaces.