Log out Go to App
Go to App

Learning Materials

Features

Discover

Chapter 8: Q27E (page 437)

Question: 27. Give an example of a closed subset\(S\)of\({\mathbb{R}^{\bf{2}}}\)such that\({\rm{conv}}\,S\)is not closed.

Short Answer

Expert verified

The set is \(S = \left\{ {\left( {x,y} \right):{x^2}{y^2} = 1,\,\,y > 0} \right\}\).

Step by step solution

01

Assume subset  \(S\) such that \({\rm{conv }}S\) is not closed

One of the possible sets is \(S = \left\{ {\left( {x,y} \right):{x^2}{y^2} = 1,\,\,y > 0} \right\}\). This set is not closed as the equation \({x^2}{y^2} = 1,\,\,\,y > 0\) is of a hyperbola in an upper half-plane that is shown below:

02

Check whether the assumed set \(S\)is in \({\mathbb{R}^{\bf{2}}}\)

The hyperbola\(xy = 1\)opens upwards; that is, it satisfies all values of \(x\) and for \(y > 0\).

So, the set \(S\) is in \({\mathbb{R}^{\bf{2}}}\).

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

In Exercises 1-6, determine if the set of points is affinely dependent. (See Practice Problem 2.) If so, construct an affine dependence relation for the points.

3.\(\left( {\begin{aligned}{{}}1\\2\\{ - 1}\end{aligned}} \right),\left( {\begin{aligned}{{}}{ - 2}\\{ - 4}\\8\end{aligned}} \right),\left( {\begin{aligned}{{}}2\\{ - 1}\\{11}\end{aligned}} \right),\left( {\begin{aligned}{{}}0\\{15}\\{ - 9}\end{aligned}} \right)\)

Repeat Exercise 25 with\({v_1} = \left[ {\begin{array}{*{20}{c}}1\\{\bf{2}}\\{ - {\bf{4}}}\end{array}} \right]\),\({v_{\bf{2}}} = \left[ {\begin{array}{*{20}{c}}{\bf{8}}\\{\bf{2}}\\{ - {\bf{5}}}\end{array}} \right]\), \({v_{\bf{3}}} = \left[ {\begin{array}{*{20}{c}}{\bf{3}}\\{{\bf{10}}}\\{ - {\bf{2}}}\end{array}} \right]\), \({\bf{a}} = \left[ {\begin{array}{*{20}{c}}{\bf{0}}\\{\bf{0}}\\{\bf{8}}\end{array}} \right]\), and \({\bf{b}} = \left[ {\begin{array}{*{20}{c}}{.{\bf{9}}}\\{{\bf{2}}.{\bf{0}}}\\{ - {\bf{3}}.{\bf{7}}}\end{array}} \right]\).

Use partitioned matrix multiplication to compute the following matrix product, which appears in the alternative formula (5) for a Bezier curve.

\(\left( {\begin{aligned}{{}}1&0&0&0\\{ - 3}&3&0&0\\3&{ - 6}&3&0\\{ - 1}&3&{ - 3}&1\end{aligned}} \right)\left( {\begin{aligned}{{}}{{{\mathop{\rm p}\nolimits} _0}}\\{{{\mathop{\rm p}\nolimits} _1}}\\{{{\mathop{\rm p}\nolimits} _2}}\\{{{\mathop{\rm p}\nolimits} _3}}\end{aligned}} \right)\)

Questions: Let \({F_{\bf{1}}}\) and \({F_{\bf{2}}}\) be 4-dimensional flats in \({\mathbb{R}^{\bf{6}}}\), and suppose that \({F_{\bf{1}}} \cap {F_{\bf{2}}} \ne \phi \). What are the possible dimension of \({F_{\bf{1}}} \cap {F_{\bf{2}}}\)?

Let \({\bf{x}}\left( t \right)\) be a cubic Bézier curve determined by points \({{\bf{p}}_o}\), \({{\bf{p}}_1}\), \({{\bf{p}}_2}\), and \({{\bf{p}}_3}\).

a. Compute the tangent vector \({\bf{x}}'\left( t \right)\). Determine how \({\bf{x}}'\left( 0 \right)\) and \({\bf{x}}'\left( 1 \right)\) are related to the control points, and give geometric descriptions of the directions of these tangent vectors. Is it possible to have \({\bf{x}}'\left( 1 \right) = 0\)?

b. Compute the second derivative and determine how and are related to the control points. Draw a figure based on Figure 10, and construct a line segment that points in the direction of . [Hint: Use \({{\bf{p}}_1}\) as the origin of the coordinate system.]
See all solutions

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Read Explanation

Decision Maths

Read Explanation

Calculus

Read Explanation

Statistics

Read Explanation

Mechanics Maths

Read Explanation

Applied Mathematics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.

Sign-up for free