Question

Let \(A\) and \(B\) be two convex sets in a normed space \(X\). Show that \(\operatorname{conv}(A \cup B)=\\{\lambda x+(1-\lambda) y ; x \in A, y \in B, \lambda \in[0,1]\\}\) Hint: Show first that the set on the right-hand side is convex.

Short answer

\( \operatorname{conv}(A \cup B) = \{ \lambda x + (1 - \lambda) y ; x \in A, y \in B, \lambda \in [0,1] \} \).

Step-by-step solution

Understand the Problem

We need to show the equality between the convex hull of the union of two convex sets and the set containing all convex combinations of elements from both sets.

Show the Right-Hand Set is Convex

To show that the set \{\lambda x + (1 - \lambda) y ; x \in A, y \in B, \lambda \in [0,1]\} is convex, pick any two elements of this set, say \(\lambda_1 x_1 + (1 - \lambda_1) y_1\) and \(\lambda_2 x_2 + (1 - \lambda_2) y_2\), and any \( \theta \in [0,1] \).

Combine Two Convex Combinations

Consider \( z = \theta ( \lambda_1 x_1 + (1 - \lambda_1) y_1 ) + (1 - \theta)( \lambda_2 x_2 + (1 - \lambda_2) y_2 ) \). Distribute \( \theta \) and \(1 - \theta\) to get: \( z = \theta \lambda_1 x_1 + \theta (1 - \lambda_1) y_1 + (1 - \theta) \lambda_2 x_2 + (1 - \theta)(1 - \lambda_2) y_2 \).

Aggregate Terms

Rewrite the expression as follows: \( z = (\theta \lambda_1 x_1 + (1 - \theta) \lambda_2 x_2) + (\theta (1 - \lambda_1) y_1 + (1 - \theta)(1 - \lambda_2) y_2) \). Notice that this expression represents a convex combination of points \( x_i \) from \( A \) and \( y_i \) from \( B \) again, confirming that the set is convex.

Show Inclusion in Convex Hull

By definition, each point \(\lambda x + (1 - \lambda) y\) belongs to the convex hull of the union of \( A \) and \( B \), thus \( \{\lambda x + (1 - \lambda) y ; x \in A, y \in B, \lambda \in [0,1]\} \subseteq \operatorname{conv}(A \cup B)\).

Show Convex Hull Inclusion

Every element of \( \operatorname{conv}(A \cup B) \) can be expressed as a convex combination of elements from \( A \cup B \). Each element in \( A \cup B \) can be represented in the form \( \lambda x + (1 - \lambda) y \) for some \(x \in A, y \in B \) and \( \lambda \in [0,1] \), ensuring \(\operatorname{conv}(A \cup B) \subseteq \{\lambda x + (1 - \lambda) y ; x \in A, y \in B, \lambda \in [0,1] \}. \)

Conclusion

Since we have both inclusions, we can conclude that \(\operatorname{conv}(A \cup B) = \{\lambda x + (1 - \lambda) y ; x \in A, y \in B, \lambda \in [0,1] \}.\)

Key concepts

convex hull

In mathematics, a convex hull is the smallest convex set that contains a given set of points.
Imagine stretching a rubber band around a set of nails on a board; the shape the rubber band forms when tightly wrapped around the nails represents the convex hull of those points.
More formally, for a set of points \(S\) in a normed space, the convex hull is the set of all convex combinations of points in \(S\).
This can be written as:
\(\text{conv}(S) = \{ \lambda_1 x_1 + \lambda_2 x_2 + ... + \lambda_n x_n \ | \ x_i \in S, \lambda_i \geq 0, \sum_{i=1}^n \lambda_i = 1 \}\).
For two convex sets \(A\) and \(B\), the convex hull of their union \(A \cup B\) is the set of all convex combinations of points taken from either \(A\) or \(B\), ensuring all possible points that can be formed by combining points from both sets with weights summing to 1 are included.

convex combination

A convex combination involves blending points from a set in a particular weighted manner such that the resulting point always lies within the outer boundary of the set.
Mathematically, if you have points \(x_1, x_2, ..., x_n\) and weights \(\lambda_1, \lambda_2, ..., \lambda_n\) with each weight \(\lambda_i\) being non-negative and summing to 1, the convex combination can be expressed as: \(\text{Convex Combination} = \lambda_1 x_1 + \lambda_2 x_2 + ... + \lambda_n x_n\).
For example, in a normed space, any point \(z\) in the set \( \{\lambda x + (1 - \lambda) y; x \in A, y \in B, \lambda \in [0,1] \}\) is a convex combination where \(x\) is from set \(A\) and \(y\) is from set \(B\).
This property is crucial in showing that the aforementioned combination represents a point within the convex hull of \(A \cup B\).

normed space

A normed space is a vector space equipped with a function called a norm, which assigns a positive length or size to all its vectors except the zero vector.
Essentially, it’s a way to measure the 'distance' in the space. The norm of a vector \(x\), denoted as \(\|x\|\), satisfies the following properties:

• \(\|x\| \geq 0\) (Non-negativity), and \(\|x\| = 0\) if and only if \(x = 0\).


• \(\| \alpha x \| = | \alpha | \| x \| \) for any scalar \(\alpha\) (Homogeneity).


• \(\| x + y \| \leq \| x \| + \| y \| \) (Triangle Inequality).

In a normed space, the distance between two points \(x\) and \(y\) is given by \(\|x - y\|\). Understanding normed spaces is fundamental for analyzing convex sets within those spaces, as the norms help in concretely defining the boundaries and measurable properties within the space.

set theory

Set theory is the mathematical study of sets, which are collections of objects.
It provides a fundamental language for modern mathematics, underpinning concepts in different areas such as logic, algebra, and topology.
Key concepts in set theory include:

• **Union**: \(A \cup B\) consists of all elements that are in \(A\), \(B\), or both.


• **Intersection**: \(A \cap B\) consists of elements that are both in \(A\) and \(B\).


• **Subsets**: A set \(A\) is a subset of a set \(B\) if all elements of \(A\) are also elements of \(B\) (denoted \(A \subseteq B\)).


• **Convex Sets**: In a normed space, a set \(A\) is convex if for any two points in \(A\), the line segment joining them is entirely within \(A\).

In the context of convex sets, set theory helps us form and manipulate sets to understand properties like the union of convex sets and the resulting convex hull.