Question

Let \(X\) be a reflexive strictly convex Banach space, and \(C\) a closed convex set in \(X\). Show that there is a unique nearest point to \(x\) in \(C\). Hint: \(x\) has a nearest point because \(X\) is reflexive (Exercise 3.104). Assume that \(x=0\) and \(\operatorname{dist}(x, C)=1\). Let \(c_{1} \neq c_{2} \in C\) satisfy \(\left\|c_{1}\right\|=\left\|c_{2}\right\|=1\). Then \(\frac{1}{2}\left(c_{1}+c_{2}\right) \in C\), yet \(\left\|\frac{1}{2}\left(c_{1}+c_{2}\right)\right\|<1\) by the strict convexity, a contradiction.

Short answer

In a reflexive strictly convex Banach space, each point has a unique nearest point in any closed convex set.

Step-by-step solution

Understand the Problem

We need to show that in a reflexive strictly convex Banach space, every point has a unique nearest point in a closed convex set.

Use the Hint

The hint states that since the Banach space is reflexive, every point has a nearest point in a closed convex set. We will use this fact.

Assume the Contrary

Assume that there are two different points, say, \(c_1\) and \(c_2\), in the closed convex set \(C\) that are both nearest to a point \(x = 0\), and \(\text{dist}(x, C) = 1\). This means \(\|c_1\| = \|c_2\| = 1\).

Consider the Midpoint

Consider the midpoint between \(c_1\) and \(c_2\): \( \frac{1}{2}(c_1 + c_2) \in C \).

Apply Strict Convexity

Using the strict convexity of the Banach space, we know that \( \left\| \frac{1}{2}(c_1 + c_2) \right\| < 1 \).

Derive a Contradiction

However, \( \frac{1}{2}(c_1 + c_2) \) should also be a nearest point to \(x\) with a norm equal to 1, contradicting the fact that its norm is less than 1.

Conclude Uniqueness

Therefore, our assumption that there are two different nearest points is incorrect. Hence, every point has a unique nearest point in the closed convex set \(C\).

Key concepts

Convex Sets

A convex set is a fundamental concept in mathematics, especially in areas like functional analysis and optimization. A set \(C\) is said to be convex if, for any two points \(a\) and \(b\) in \(C\), the line segment connecting \(a\) and \(b\) is also entirely within \(C\). Formally, this can be expressed as:
\forall a, b oteq C\text{ and } \forall \theta\text{ with }0 ot\textless \theta ot\textless 1, \theta a + (1-\theta) b oteq C.
Understanding convexity helps in analyzing various properties of sets and functions defined on these sets. Convex sets often simplify optimization problems as they avoid local minima that are not global minima.

Nearest Point

In the context of Banach spaces, finding the nearest point in a closed convex set \(C\) to a given point \(x\) is a common problem. The nearest point to \(x\) in \(C\) is the point \(y oteq C\) that minimizes the distance \'d(x, y)\.
For any two points \(p\) and \(q\) in a metric space, the distance is usually given by the norm \(\text{\textbardbl}p - q\text{\textbardbl}\), where \(\text{\textbardbl} \text{\textbardbl}\) denotes the norm. The goal is to find a unique point \(y\) such that \(\text{\textbardbl}x - y\text{\textbardbl}\) is minimized and this point is within the set \(C\).

Strict Convexity

A Banach space is said to be strictly convex if every point on the line segment connecting any two distinct points in its space lies inside the unit ball, except the endpoints. In other words, if \(p\) and \(q\) are in the space and \(p ot\textbf{≠} q\), then for every \(0 ot\textless \theta ot\textless 1\), we have
$$ \text{\textbardbl}\theta p + (1-\theta) q\text{\textbardbl} ot\textless max(\text{\textbardbl}p\text{\textbardbl}, \text{\textbardbl}q\text{\textbardbl}) $$.
Strict convexity ensures the uniqueness of nearest points, as demonstrated in the exercise solution: if there were two nearest points, their midpoint would have a smaller norm, contradicting minimality.

Reflexivity

A Banach space is reflexive if it is isometrically isomorphic to its double dual. This means that every continuous linear functional on its dual has a representation in the space itself. Reflexivity has important consequences in functional analysis, such as the weak compactness of closed unit balls.
Reflexive Banach spaces are characterized by every point having a nearest point in any closed convex set, a property that was crucial for proving uniqueness in the given exercise.

Banach Spaces

A Banach space is a complete normed vector space. Completeness means that every Cauchy sequence in the space converges to a limit that is within the space. Banach spaces generalize many properties of finite-dimensional Euclidean spaces to infinite dimensions.
These spaces provide the framework for functional analysis, with applications across mathematics, including solving differential and integral equations. Key properties like strict convexity and reflexivity play a significant role in various theorems, as seen in the exercise about unique nearest points in closed convex sets.