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Chapter 8: Q11E (page 437)

Question: Find an example of a closed convex set S in \({\mathbb{R}^{\bf{2}}}\) such that its profile P is nonempty but \({\bf{conv}}\,P \ne S\).

Short Answer

Expert verified

A rectangle without lower and right boundaries represents a bounded convex set S in \({\mathbb{R}^2}\) such that profile P is nonempty but \({\rm{conv}}\,\,P \ne S\).

Step by step solution

01

Find the example of a closed convex set

One example of such a closed convex set is a rectangle that has all parts except lower and right boundary as shown below:

02

Create the convex hull

The convex hull can be created by outlining a triangular region with three of its vertices.

It represents one example of a bounded convexset S in \({\mathbb{R}^2}\) such that its profile P in non empty but \({\rm{conv}}\,\,P \ne S\).

So, a rectangle without lower and right boundary represents a bounded convex set S in \({\mathbb{R}^2}\) such that profile P is nonempty but \({\rm{conv}}\,\,P \ne S\).

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Most popular questions from this chapter

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

11. a. The set of all affine combinations of points in a set \(S\) is called the affine hull of \(S\).

b. If \(\left\{ {{{\rm{b}}_{\rm{1}}}{\rm{,}}.......{{\rm{b}}_{\rm{2}}}} \right\}\) is a linearly independent subset of \({\mathbb{R}^n}\) and if \({\bf{p}}\) is a linear combination of \({{\rm{b}}_{\rm{1}}}.......{{\rm{b}}_{\rm{k}}}\), then \({\rm{p}}\) is an affine combination of \({{\rm{b}}_{\rm{1}}}.......{{\rm{b}}_{\rm{k}}}\).

c. The affine hull of two distinct points is called a line.

d. A flat is a subspace.

e. A plane in \({\mathbb{R}^3}\) is a hyper plane.

Question: 17. Choose a set \(S\) of three points such that aff \(S\) is the plane in \({\mathbb{R}^3}\) whose equation is \({x_3} = 5\). Justify your work.

Use only the definition of affine dependence to show that anindexed set \(\left\{ {{v_1},{v_2}} \right\}\) in \({\mathbb{R}^{\bf{n}}}\) is affinely dependent if and only if \({v_1} = {v_2}\).

Question: 2. Let Lbe the line in \({\mathbb{R}^{\bf{2}}}\) through the points \(\left( {\begin{array}{*{20}{c}}{\bf{1}}\\{\bf{4}}\end{array}} \right)\) and \(\left( {\begin{array}{*{20}{c}}{ - {\bf{2}}}\\{ - {\bf{1}}}\end{array}} \right)\). Find a linear functional f and a real number d such that \(L = \left( {f:d} \right)\).

Question: 13. Suppose \(\left\{ {{{\rm{v}}_{\rm{1}}}{\rm{,}}{{\rm{v}}_{\rm{2}}}{\rm{,}}{{\rm{v}}_{\rm{3}}}} \right\}\) is a basis for \({\mathbb{R}^3}\). Show that Span \(\left\{ {{{\rm{v}}_{\rm{2}}} - {{\rm{v}}_{\rm{1}}},{{\rm{v}}_{\rm{3}}} - {{\rm{v}}_{\rm{1}}}} \right\}\) is a plane in \({\mathbb{R}^3}\). (Hint: What can you say about \({\rm{u}}\) and \({\rm{v}}\)when Span \(\left\{ {{\rm{u,v}}} \right\}\) is a plane?)
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