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Chapter 8: Q30E (page 437)

Question: 30. Prove that the convex hull of a bounded set is bounded.

Short Answer

Expert verified

It is shown that the convex hull of a bounded set is bounded.

Step by step solution

01

Assume a bounded set

Assume a bounded set \(S\). This implies that for the set \(S\)to lie in a convex open ball \(B\left( {0,\delta } \right)\), \(\delta \) must be positive.

02

Use theorem 9

According to theorem 9, the convex combination of unbounded set \(S\) must lie in an open ball set \(B\left( {{\rm{p}},\delta } \right)\) such that \({\rm{conv}}\,S\) is also bounded.

Thus, it shows that the convex hull of a bounded set is bounded.

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Most popular questions from this chapter

Let\({v_1} = \left[ {\begin{array}{*{20}{c}}{ - 1}\\2\end{array}} \right]\),\({v_{\bf{2}}} = \left[ {\begin{array}{*{20}{c}}{\bf{0}}\\{\bf{4}}\end{array}} \right]\),\({v_{\bf{3}}} = \left[ {\begin{array}{*{20}{c}}{\bf{2}}\\{\bf{0}}\end{array}} \right]\), and let\(S = \left\{ {{v_1},{v_2},{v_3}} \right\}\).

  1. Show that the set is affinely independent.
  2. Find the barycentric coordinates of\({p_1} = \left[ {\begin{array}{*{20}{c}}2\\3\end{array}} \right]\),\({p_{\bf{2}}} = \left[ {\begin{array}{*{20}{c}}{\bf{1}}\\{\bf{2}}\end{array}} \right]\),\({p_{\bf{3}}} = \left[ {\begin{array}{*{20}{c}}{ - 2}\\{\bf{1}}\end{array}} \right]\),\({p_{\bf{4}}} = \left[ {\begin{array}{*{20}{c}}{\bf{1}}\\{ - {\bf{1}}}\end{array}} \right]\), and\({p_{\bf{5}}} = \left[ {\begin{array}{*{20}{c}}{\bf{1}}\\{\bf{1}}\end{array}} \right]\), with respect to S.
  3. Let\(T\)be the triangle with vertices\({v_1}\),\({v_{\bf{2}}}\), and\({v_{\bf{3}}}\). When the sides of\(T\)are extended, the lines divide\({\mathbb{R}^{\bf{2}}}\)into seven regions. See Figure 8. Note the signs of the barycentric coordinates of the points in each region. For example,\({{\bf{p}}_{\bf{5}}}\)is inside the triangle\(T\)and all its barycentric coordinates are positive. Point\({{\bf{p}}_{\bf{1}}}\)has coordinates\(\left( { - , + , + } \right)\). Its third coordinate is positive because\({{\bf{p}}_{\bf{1}}}\)is on the\({{\bf{v}}_{\bf{3}}}\)side of the line through\({{\bf{v}}_{\bf{1}}}\)and\({{\bf{v}}_{\bf{2}}}\). Its first coordinate is negative because\({{\bf{p}}_{\bf{1}}}\)is opposite the\({{\bf{v}}_{\bf{1}}}\)side of the line through\({{\bf{v}}_{\bf{2}}}\)and\({{\bf{v}}_{\bf{3}}}\). Point\({{\bf{p}}_{\bf{2}}}\)is on the\({{\bf{v}}_{\bf{2}}}{{\bf{v}}_{\bf{3}}}\)edge of\(T\). Its coordinates are\(\left( {0, + , + } \right)\). Without calculating the actual values, determine the signs of the barycentric coordinates of points\({{\bf{p}}_{\bf{6}}}\),\({{\bf{p}}_{\bf{7}}}\), and\({{\bf{p}}_{\bf{8}}}\)as shown in Figure 8.

Question: Suppose that the solutions of an equation \(A{\bf{x}} = {\bf{b}}\) are all of the form \({\bf{x}} = {x_{\bf{3}}}{\bf{u}} + {\bf{p}}\), where \({\bf{u}} = \left( {\begin{array}{*{20}{c}}{\bf{5}}\\{\bf{1}}\\{ - {\bf{2}}}\end{array}} \right)\) and \({\bf{p}} = \left( {\begin{array}{*{20}{c}}{\bf{1}}\\{ - {\bf{3}}}\\{\bf{4}}\end{array}} \right)\). Find points \({{\bf{v}}_{\bf{1}}}\) and \({{\bf{v}}_{\bf{2}}}\) such that the solution set of \(A{\bf{x}} = {\bf{b}}\) is \({\bf{aff}}\left\{ {{{\bf{v}}_{\bf{1}}},\,{{\bf{v}}_{\bf{2}}}} \right\}\).

In Exercises 21-26, prove the given statement about subsets A and B of \({\mathbb{R}^n}\), or provide the required example in \({\mathbb{R}^2}\). A proof for an exercise may use results from earlier exercises (as well as theorems already available in the text).

25. \({\mathop{\rm aff}\nolimits} \left( {A \cap B} \right) \subset \left( {{\mathop{\rm aff}\nolimits} A \cap {\mathop{\rm aff}\nolimits} B} \right)\)

Question: In Exercises 21 and 22, mark each statement True or False. Justify each answer.

22 a. If \(d\)is a real number and \(f\) is a nonzero linear functional defined on \({\mathbb{R}^n}\) , then \(f:d\)is a hyperplane in \({\mathbb{R}^n}\) .

b. Given any vector n and any real number \(d\), the set \(\left\{ {x:n \cdot x = d} \right\}\) is a hyperplane.

c. If \(A\) and \(B\) are nonempty disjoint sets such that \(A\) is compact and \(B\) is closed, then there exists a hyperplane that strictly separates \(A\) and \(B\).

d. If there exists a hyperplane \(H\) such that \(H\) does not strictly separate two sets \(A\) and \(B\), then \(\left( {{\rm{conv}}\,A} \right) \cap \left( {{\rm{conv}}\,B} \right) \ne \emptyset \) and \(B\).


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.]
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