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Chapter 8: Q19E (page 437)

Question:19.In Exercises 17鈥�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 11 and 12, mark each statement True or False. Justify each answer.

12.a. The essential properties of Bezier curves are preserved under the action of linear transformations, but not translations.

b. When two Bezier curves \({\mathop{\rm x}\nolimits} \left( t \right)\) and \(y\left( t \right)\) are joined at the point where \({\mathop{\rm x}\nolimits} \left( 1 \right) = y\left( 0 \right)\), the combined curve has \({G^0}\) continuity at that point.

c. The Bezier basis matrix is a matrix whose columns are the control points of the curve.

In Exercises 1-4, write y as an affine combination of the other point listed, if possible.

\({{\bf{v}}_{\bf{1}}} = \left( {\begin{aligned}{*{20}{c}}{\bf{1}}\\{\bf{2}}\\{\bf{0}}\end{aligned}} \right)\), \({{\bf{v}}_{\bf{2}}} = \left( {\begin{aligned}{*{20}{c}}{\bf{2}}\\{ - {\bf{6}}}\\{\bf{7}}\end{aligned}} \right)\), \({{\bf{v}}_{\bf{3}}} = \left( {\begin{aligned}{*{20}{c}}{\bf{4}}\\{\bf{3}}\\{\bf{1}}\end{aligned}} \right)\), \({\bf{y}} = \left( {\begin{aligned}{*{20}{c}}{ - {\bf{3}}}\\{\bf{4}}\\{ - {\bf{4}}}\end{aligned}} \right)\)

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 \({F_{\bf{1}}} \cap {F_{\bf{2}}}\)?

Question: 26. Let \({\rm{q}} = \left( \begin{array}{l}2\\3\end{array} \right)\), \({\rm{p}} = \left( \begin{array}{l}6\\1\end{array} \right)\). Find a hyperplane \(\left( {f:d} \right)\) that strictly separates \(B\left( {{\rm{q}},3} \right)\) and \(B\left( {{\rm{p}},1} \right)\).

Question: In Exercises 5-8, find the minimal representation of the polytope defined by the inequalities \(A{\bf{x}} \le {\bf{b}}\) and \({\bf{x}} \ge {\bf{0}}\).

5. \(A = \left( {\begin{array}{*{20}{c}}1&2\\3&1\end{array}} \right),{\rm{ }}{\bf{b}} = \left( {\begin{array}{*{20}{c}}{{\bf{10}}}\\{{\bf{15}}}\end{array}} \right)\)
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