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Chapter 8: Q18E (page 437)

Question: Let v be an element of the convex set S. prove that v is an extreme point of S if an only if the set \(\left\{ {{\bf{x}} \in S:{\bf{x}} \ne {\bf{v}}} \right\}\) is convex.

Short Answer

Expert verified

The point v is an extreme point of S if and only if the set \(\left\{ {{\bf{x}} \in S:\,\,{\bf{x}} \ne {\bf{v}}} \right\}\) is convex.

Step by step solution

01

Check for T  (in S), which has elements y and z

Let y and z are in T, then \(\overline {yz} \subseteq S\), since S is convex.

If v is an extreme point of S, \(v \notin yz\), so \(\overline {yz} \subset T\).

So, T is convex.

02

Check whether the given set is convex or not

Let \({\bf{v}} \in S\), but not an extreme point of S. Then there are y and z in S such that \(v \in \overline {yz} \) with \(v \ne y\) and \(v \ne z\). It proves that T is not convex.

So, v is an extreme point of S if and only if the set \(\left\{ {{\bf{x}} \in S:\,\,{\bf{x}} \ne {\bf{v}}} \right\}\) is convex.

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Most popular questions from this chapter

In Exercises 21–24, a, b, and c are noncollinear 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.

21. Show that the area of\(\Delta {\bf{abc}}\)is\(det\left[ {\begin{array}{*{20}{c}}{\overrightarrow {\bf{a}} }&{\overrightarrow {\bf{b}} }&{\overrightarrow {\bf{c}} }\end{array}} \right]/2\).

[Hint:Consult Sections 3.2 and 3.3, including the Exercises.]

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 \(B\).

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.

Question: 24. Repeat Exercise 23 for \({v_1} = \left( \begin{array}{l}1\\2\end{array} \right)\), \({v_2} = \left( \begin{array}{l}5\\1\end{array} \right)\), \({v_3} = \left( \begin{array}{l}4\\4\end{array} \right)\) and \(p = \left( \begin{array}{l}2\\3\end{array} \right)\).

Question: 16. Let \({\rm{v}} \in {\mathbb{R}^n}\)and let \(k \in \mathbb{R}\). Prove that \(S = \left\{ {{\rm{x}} \in {\mathbb{R}^n}:{\rm{x}} \cdot {\rm{v}} = k} \right\}\)is an affine subset of \({\mathbb{R}^n}\).
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