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Chapter 8: Problem 1

In \(\mathbb{R}^{2},\) let \(S=\left\\{\left[\begin{array}{l}{0} \\\ {y}\end{array}\right] : 0 \leq y<1\right\\} \cup\left\\{\left[\begin{array}{l}{2} \\ {0}\end{array}\right]\right\\} .\) Describe (or sketch) the convex hull of \(S .\)

Short Answer

Expert verified
The convex hull is a triangle with vertices at \((0, 0)\), \((2, 0)\), and just below \((0, 1)\).

Step by step solution

01

Understanding the Set S

The set \(S\) in \(\mathbb{R}^2\) consists of two parts: all vectors of the form \(\begin{bmatrix} 0 \ y \end{bmatrix}\) where \(0 \leq y < 1\); and the vector \(\begin{bmatrix} 2 \ 0 \end{bmatrix}\). The first part forms a vertical line segment along the y-axis from \((0, 0)\) to \((0, 1)\) but not including the point \((0, 1)\). The second part is a single point at \((2, 0)\).
02

Visualizing S and Finding Extremal Points

Visualize the components of \(S\): a line segment from \((0, 0)\) to just below \((0, 1)\), and a point at \((2, 0)\). The extremal points of \(S\) that will help form the convex hull are \((0, 0)\), just below \((0, 1)\), and \((2, 0)\).
03

Connecting Extremal Points to Form the Convex Hull

The convex hull of a set is the smallest convex set that contains all points in \(S\). To form the convex hull, connect the extremal point \((0, 0)\) to just below \((0, 1)\) creating a vertical boundary; also connect \((0, 0)\) to \((2, 0)\) creating a base; finally, connect just below \((0, 1)\) directly to \((2, 0)\), creating a slanted boundary.
04

Describe the Convex Hull

The convex hull of \(S\) is a triangle with vertices at \((0, 0)\), just below \((0, 1)\), and \((2, 0)\). The area within these vertices, including the boundary line from \((0, 0)\) to \((0, 1)\), but excluding the top boundary itself, forms a convex region.

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Key Concepts

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

Extremal Points
In geometry, extremal points play a crucial role in understanding the framework of a convex hull. An extremal point is essentially a 'corner' of a shape that contributes to the boundary of the convex region. In our exercise, we consider the set \(S\) in \(\mathbb{R}^2\). This involves a line segment extending from \((0, 0)\) to an endpoint just before \((0, 1)\), and a distinct point at \((2, 0)\). These extremal points include the starting points of the line, its ending, and the lone point far right.
  • \((0, 0)\): The base point on the y-axis.
  • Just below \((0, 1)\): The top point on the y-axis, just before reaching 1.
  • \((2, 0)\): A separate point on the x-axis.
Understanding these points helps us visualize and create the boundary of our convex hull. They serve as essential vertices that connect up to form a minimal enclosing 'shell' around the set.
Convex Set
A convex set is a collection of points where any line segment drawn between any two points within the set remains entirely inside the set. This essential property distinguishes convex sets from other geometric configurations.
In the context of the convex hull of set \(S\), it means that any point lying on or within the boundaries that connect
  • the line from \((0, 0)\) to just below \((0, 1)\),
  • the horizontal base from \((0, 0)\) to \((2, 0)\), and
  • the slant from just below \((0, 1)\) to \((2, 0)\)
will also be part of the convex hull. The concept of a convex set ensures a predictable, unbroken boundary that plays a foundational role in geometric, and computational, analyses.
Visualization of Sets
Visualizing a set helps immensely in predicting and understanding the outcome of constructing a convex hull. For the set \(S\), imagine first the line segment extending vertically up the y-axis from \((0, 0)\) almost to \((0, 1)\). Then, envision the isolated point at \((2, 0)\).
To further aid visualization, trace lines connecting these extremal points: from \((0, 0)\) to just below \((0, 1)\), from \((0, 0)\) across to \((2, 0)\), and finally, from the near-top of the y-axis diagonally to \((2, 0)\).
  • Imagine this forming a triangular shape.
  • Think about how the space within and beneath those lines forms a completed region.
Visual aids like sketches can enrich one's comprehension by emphasizing how these lines create a cohesive convex region that encompasses all points within set \(S\). This graphical representation guides us in deducing properties and applications related to convex sets in multi-dimensional spaces.

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Most popular questions from this chapter

Let \(\mathbf{v}_{1}=\left[\begin{array}{r}{1} \\ {3} \\\ {-6}\end{array}\right], \mathbf{v}_{2}=\left[\begin{array}{r}{7} \\ {3} \\\ {-5}\end{array}\right], \mathbf{v}_{3}=\left[\begin{array}{r}{3} \\ {9} \\\ {-2}\end{array}\right], \mathbf{a}=\left[\begin{array}{l}{0} \\ {0} \\\ {9}\end{array}\right]\) \(\mathbf{b}=\left[\begin{array}{r}{1.4} \\ {1.5} \\\ {-3.1}\end{array}\right],\) and \(\mathbf{x (t)=\mathbf{a}+t \mathbf{b}\) for \(t \geq 0 .\) Find the point where the ray \(\mathbf{x}(t)\) intersects the plane that contains the triangle with vertices \(\mathbf{v}_{1}, \mathbf{v}_{2},\) and \(\mathbf{v}_{3} .\) Is this point inside the triangle?

In Exercises \(15-20,\) write a formula for a linear functional \(f\) and specify a number \(d,\) so that \([f : d]\) is the hyperplane \(H\) described in the exercise. Let \(H\) be the column space of the matrix \(B=\left[\begin{array}{rr}{1} & {0} \\\ {5} & {2} \\ {-4} & {-4}\end{array}\right]\) That is, \(H=\operatorname{Col} B\)

Let \(\mathbf{p}_{1}=\left[\begin{array}{r}{2} \\ {-3} \\ {1} \\\ {2}\end{array}\right], \quad \mathbf{p}_{2}=\left[\begin{array}{r}{1} \\ {2} \\\ {-1} \\ {3}\end{array}\right], \quad \mathbf{n}_{1}=\left[\begin{array}{l}{1} \\ {2} \\ {4} \\\ {2}\end{array}\right], \quad\) and \(\mathbf{n}_{2}=\left[\begin{array}{l}{2} \\\ {3} \\ {1} \\ {5}\end{array}\right] ;\) let \(H_{1}\) be the hyperplane in \(\mathbb{R}^{4}\) through \(\mathbf{p}_{1}\) with normal \(\mathbf{n}_{1} ;\) and let \(H_{2}\) be the hyperplane through \(\mathbf{p}_{2}\) with normal \(\mathbf{n}_{2} .\) Give an explicit description of \(H_{1} \cap H_{2}\) . [Hint: Find a point \(\mathbf{p}\) in \(H_{1} \cap H_{2}\) and two linearly independent vectors \(\mathbf{v}_{1}\) and \(\mathbf{v}_{2}\) that span a subspace parallel to the \(2-\) dimensional flat \(H_{1} \cap H_{2} . ]\)

Mark each statement True or False. Justify each answer. a. If \(\mathbf{y}=c_{1} \mathbf{v}_{1}+c_{2} \mathbf{v}_{2}+c_{3} \mathbf{v}_{3}\) and \(c_{1}+c_{2}+c_{3}=1,\) then \(\mathbf{y}\) is a convex combination of \(\mathbf{v}_{1}, \mathbf{v}_{2},\) and \(\mathbf{v}_{3}\) . b. If \(S\) is a nonempty set, then conv \(S\) contains some points that are not in \(S\) . c. If \(S\) and \(T\) are convex sets, then \(S \cup T\) is also convex.

A polyhedron \((3-\text { polytope) is called regular if all its facets }\) are congruent regular polygons and all the angles at the vertices are equal. Supply the details in the following proof that there are only five regular polyhedra. a. Suppose that a regular polyhedron has \(r\) facets, each of which is a \(k\) -sided regular polygon, and that \(s\) edges meet at each vertex. Letting \(v\) and \(e\) denote the numbers of vertices and edges in the polyhedron, explain why \(k r=2 e\) and \(s v=2 e\) b. Use Euler's formula to show that \(\frac{1}{s}+\frac{1}{k}=\frac{1}{2}+\frac{1}{e}\) c. Find all the integral solutions of the equation in part (b) that satisfy the geometric constraints of the problem. (How small can \(k\) and \(s\) be? \()\)
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