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Chapter 1: Problem 93

For three points \(P, Q\), and \(R\) in \(\mathbb{R}^3\) (or, more generally, in \(\mathbb{R}^n\) ) we say that \(R\) is between \(P\) and \(Q\) if \(R\) is on the line segment connecting \(P\) and \(Q\) ( \(R=P\) and \(R=Q\) are allowed). A subset \(A\) of \(\mathbb{R}^3\) is called convex if for any two points \(P\) and \(Q\) in \(A\), every point \(R\) which is between \(P\) and \(Q\) is also in \(A\). For instance, an ellipsoid is convex, a banana is not. Now for the problem: Suppose \(A\) and \(B\) are convex subsets of \(\mathbb{R}^3\). Let \(C\) be the set of all points \(R\) for which there are points \(P\) in \(A\) and \(Q\) in \(B\) such that \(R\) lies between \(P\) and \(Q\). Does \(C\) have to be convex?

Short Answer

Expert verified
Yes, \C\ has to be convex.

Step by step solution

01

Understand the Definition of Convex Sets

A set in \(\backslashmathbb{R}^3\) is called convex if for any two points \(P\) and \(Q\) in the set, every point \(R\) that lies on the line segment connecting \(P\) and \(Q\) is also in the set. In simpler terms, a set is convex if you can connect any two points within the set with a straight line and every point on that line is also inside the set.
02

Define C Based on Sets A and B

Given two convex subsets \(A\) and \(B\) of \(\backslashmathbb{R}^3\), define the set \(C\). The set \(C\) consists of all points \(R\) such that there exist points \(P\) in \A\ and \(Q\) in \B\ with \(R\) lying between \(P\) and \(Q\). Mathematically, R lies on the line segment that connects \(P\) and \(Q\).
03

Check If C is Convex

To check if \(C\) is convex, take any two points \R_1 \text{ and } R_2\ in \(C\). We need to show that the line segment between \(R_1\) and \(R_2\) lies entirely within \(C\). Since \(R_1\) is in \(C\), there exist points \P_1 \text{ in } A\ and \Q_1 \text{ in } B\ with \(R_1\) lying between \P_1\ and \Q_1\. Similarly, since \(R_2\) is in \(C\), there exist points \P_2 \text{ in } A\ and \Q_2 \text{ in } B\ with \(R_2\) lying between \P_2\ and \Q_2\.
04

Analyze Line Segments

Consider any point \(R\) on the line segment between \R_1\ and \R_2\. Since \(R_1\) lies between \P_1\ and \Q_1\ and \(R_2\) lies between \P_2\ and \Q_2\, any point \(R\) can be represented as a convex combination of these points, say \R \text{ in terms of } R_1 \text{ and } R_2\.
05

Conclude the Convexity of C

Since \(A\) and \(B\) are convex, all points on the line segment from \(P_1\) to \(P_2\) (which lie in \(A\)) and the line segment from \Q_1\ to \Q_2\ (which lie in \(B\)) are in \(A\) and \(B\) respectively. Hence, any point \(R\) on the line segment between \R_1\ and \R_2\ also lies within \(C\), demonstrating that \(C\) is convex.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Convexity in Mathematics
Convexity is a key concept in mathematics, closely tied to geometry and linear algebra. A set in \(\backslashmathbb{R}^3\) is termed convex if, for any two points \(P\) and \(Q\) within the set, the entire line segment connecting them is also contained within the set. Consider an ellipsoid or a cube; draw a line between any two points inside them, and every point on this line stays inside the shape, making it convex. Conversely, a shape like a banana, which curves inward, is not convex.
This fundamental property of convexity helps simplify many problems in optimization, economics, and other fields. It ensures that local properties hold globally within these sets, making calculations and predictions easier to manage.
Line Segments in 3D Space
In 3D geometry, a line segment is defined by two endpoints, such as points \(P\) and \(Q\). This segment includes every point \(R\) that lies directly between \(P\) and \(Q\). Mathematically, \(R\) can be represented as a convex combination of \(P\) and \(Q\). If \(t \in [0, 1]\), then any point on the line segment can be written as \(R = tP + (1-t)Q\). This combination assures that \(R\) stays on the line defined by \(P\) and \(Q\), facilitating various geometric analyses.
This concept extends to scenarios where we analyze subsets of 3D space, ensuring that the properties like convexity hold even when connecting points from different subsets, as explored further in the problem involving sets \(A\), \(B\), and \(C\).
Geometric Properties
Geometric properties like convexity and the behavior of line segments in 3D space are crucial for understanding complex shapes and structures. These properties ensure that when connecting any two points within specific sets (such as convex sets \(A\) and \(B\)), the resultant set \(C\) retains the same properties.
In the provided problem, analyzing convex sets \(A\) and \(B\), and defining set \(C\) as the collection of all points \(R\) lying between any points in \(A\) and \(B\), we demonstrate how maintaining convexity ensures that any line segment connecting points in \(C\) remains within \(C\). This intrinsic property ultimately simplifies analyses and validations of geometric shapes and offers powerful insights in various mathematical applications.

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