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Chapter 8: Q19E (page 437)

Question:19.In Exercises 17–20, prove the given statement about subsets\(A\)and\(B\)of\({\mathbb{R}^n}\). A proof for an exercise may use results of earlier exercises.

19. a. \(\left( {\left( {{\rm{conv}}\,A} \right) \cup \left( {{\rm{conv}}\,B} \right)} \right) \subset {\rm{conv}}\left( {A \cup B} \right)\),

b. Find an example in\({\mathbb{R}^2}\)to show that equality need not hold in part

(a)

Short Answer

Expert verified
  1. It is shown that \(\left( {{\rm{conv}}\,A\,\, \cup {\rm{conv}}\,B} \right) \subset {\rm{conv}}\,\left( {{\rm{A}} \cup \,B} \right)\).
  2. \(A\) be the two adjacent corners of the square, and B be the other two corners of the square.

Step by step solution

01

(a) Step 1:Use the result of Exercise 18

It is known that\({\rm{conv}}\,A \subset {\rm{conv}}\,B\). Moreover, the combination of \(A\) or \(B\)must contains all the combinations of A.

Similarly, convex combinations of \(A\) must contain every convex combination of \(A\) or \(B\); that is,\({\rm{conv}}\,A \subset {\rm{conv}}\,\left( {{\rm{A}} \cup \,B} \right)\).

02

Step 2:Draw a conclusion

If \({\rm{conv}}\,A \subset {\rm{conv}}\,\left( {{\rm{A}} \cup \,B} \right)\) is true, then \(\left( {\left( {conv\,A} \right)U\left( {conv\,B} \right)} \right) \subset conv\left( {AUB} \right)\) must also be true as a convex combination of points ofA orconvex combination of points of \(B\), must contain some or all points of convex combinations of \(A\).

Thus,\(\left( {\left( {conv\,A} \right)U\left( {conv\,B} \right)} \right) \subset conv\left( {AUB} \right)\).

03

(b) Step 3:  Assume an example in \({\mathbb{R}^2}\) as a requirement

Consider a square and assume \(A\) be the two adjacent corners of the square, whereas B be the other two corners of the square.

Then, \({\rm{conv}}\,A\,\, \cup {\rm{conv}}\,B\) is a set of convex of \(A\) or convex of \(B\), which represents the opposite sides of the square, whereas \({\rm{conv}}\,\left( {A\,\, \cup \,B} \right)\) is the convex combination of points of \(A\) or \(B\) which represents the opposite sides of the perimeter of the square.

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Most popular questions from this chapter

In Exercises 21–24, a, b, and c are non-collinear points in\({\mathbb{R}^{\bf{2}}}\)and p is any other point in\({\mathbb{R}^{\bf{2}}}\). Let\(\Delta {\bf{abc}}\)denote the closed triangular region determined by a, b, and c, and let\(\Delta {\bf{pbc}}\)be the region determined by p, b, and c. For convenience, assume that a, b, and c are arranged so that\(\left[ {\begin{array}{*{20}{c}}{\overrightarrow {\bf{a}} }&{\overrightarrow {\bf{b}} }&{\overrightarrow {\bf{c}} }\end{array}} \right]\)is positive, where\(\overrightarrow {\bf{a}} \),\(\overrightarrow {\bf{b}} \) and\(\overrightarrow {\bf{c}} \)are the standard homogeneous forms for the points.

22. Let p be a point on the line through a and b. Show that\(det\left[ {\begin{array}{*{20}{c}}{\overrightarrow {\bf{a}} }&{\overrightarrow {\bf{b}} }&{\overrightarrow {\bf{p}} }\end{array}} \right] = 0\).

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.

5.\(\left( {\begin{aligned}{{}}1\\0\\{ - 2}\end{aligned}} \right),\left( {\begin{aligned}{{}}0\\1\\1\end{aligned}} \right),\left( {\begin{aligned}{{}}{ - 1}\\5\\1\end{aligned}} \right),\left( {\begin{aligned}{{}}0\\5\\{ - 3}\end{aligned}} \right)\)

Question: 19. Let \(S\) be an affine subset of \({\mathbb{R}^n}\) , suppose \(f:{\mathbb{R}^n} \to {\mathbb{R}^m}\)is a linear transformation, and let \(f\left( S \right)\) denote the set of images \(\left\{ {f\left( {\rm{x}} \right):{\rm{x}} \in S} \right\}\). Prove that \(f\left( S \right)\)is an affine subset of \({\mathbb{R}^m}\).


Prove Theorem 6 for an affinely independent set\(S = \left\{ {{v_1},...,{v_k}} \right\}\)in\({\mathbb{R}^{\bf{n}}}\). [Hint:One method is to mimic the proof of Theorem 7 in Section 4.4.]

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