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Chapter 8: Q28E (page 437)

Question:28. Give an example of a compact set\(A\)and a closed set\(B\)in\({\mathbb{R}^2}\)such that\(\left( {{\rm{conv}}\,A} \right) \cap \left( {{\rm{conv}}\,B} \right) = \emptyset \)but\(A\)and\(B\)cannot be strictly separated by a hyperplane.

Short Answer

Expert verified

The sets are \(A = \left\{ {\left( {x,y} \right):\left| x \right| < 1,\,\,y = 0} \right\}\), and \(B = \left\{ {\left( {x,y} \right):{x^2}{y^2} = 1,\,\,y > 0} \right\}\).

Step by step solution

01

Assume set \(A\)and \(B\) as per required condition

One of the possible sets is \(B = \left\{ {\left( {x,y} \right):{x^2}{y^2} = 1,\,\,y > 0} \right\}\) and \(A = \left\{ {\left( {x,y} \right):\left| x \right| < 1,\,\,y = 0} \right\}\). This set is in \({\mathbb{R}^2}\).

02

Check whether \(\left( {{\rm{conv}}\,A} \right) \cap \left( {{\rm{conv}}\,B} \right) = \emptyset \)

The convex combination of both sets is null as the set\(B\)opens only in half-plane, whereas the set\(A\)lies within\( - 1 < x < 1\).

Thus, satisfy the condition\(\left( {{\rm{conv}}\,A} \right) \cap \left( {{\rm{conv}}\,B} \right) = \emptyset \).

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Most popular questions from this chapter

The “B” in B-spline refers to the fact that a segment \({\bf{x}}\left( t \right)\)may be written in terms of a basis matrix, \(\,{M_S}\) , in a form similar to a Bézier curve. That is,

\({\bf{x}}\left( t \right) = G{M_S}{\bf{u}}\left( t \right)\)for \(\,0 \le t \le 1\)

where \(G\) is the geometry matrix \(\,\left( {{{\bf{p}}_{\bf{0}}}\,\,\,\,{{\bf{p}}_{{\bf{1}}\,\,\,}}\,{{\bf{p}}_{\bf{2}}}\,\,\,{{\bf{p}}_{\bf{3}}}} \right)\)and \({\bf{u}}\left( {\bf{t}} \right)\) is the column vector \(\left( {1,\,\,t,\,\,{t^2},\,{t^3}} \right)\) . In a uniform B-spline, each segment uses the same basis matrix \(\,{M_S}\), but the geometry matrix changes. Construct the basis matrix \(\,{M_S}\) for \({\bf{x}}\left( t \right)\).

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.

2.\(\left( {\begin{aligned}{{}}2\\1\end{aligned}} \right),\left( {\begin{aligned}{{}}5\\4\end{aligned}} \right),\left( {\begin{aligned}{{}}{ - 3}\\{ - 2}\end{aligned}} \right)\)

Question: 25. Let \(p = \left( \begin{array}{l}1\\1\end{array} \right)\). Find a hyperplane \(\left( {f:d} \right)\) that strictly separates \(B\left( {0,3} \right)\) and \(B\left( {p,1} \right)\). (Hint: After finding \(f\), show that the point \(v = \left( {1 - .75} \right)0 + .75p\) is neither in \(B\left( {0,3} \right)\) nor in \(B\left( {p,1} \right)\)).

Let \({\bf{x}}\left( t \right)\) be a B-spline in Exercise 2, with control points \({{\bf{p}}_o}\), \({{\bf{p}}_1}\) , \({{\bf{p}}_2}\) , and \({{\bf{p}}_3}\).

a. Compute the tangent vector \({\bf{x}}'\left( t \right)\) and determine how the derivatives \({\bf{x}}'\left( 0 \right)\) and \({\bf{x}}'\left( 1 \right)\) are related to the control points. Give geometric descriptions of the directions of these tangent vectors. Explore what happens when both \({\bf{x}}'\left( 0 \right)\)and \({\bf{x}}'\left( 1 \right)\)equal 0. Justify your assertions.

b. Compute the second derivative and determine how and are related to the control points. Draw a figure based on Figure 10, and construct a line segment that points in the direction of . [Hint: Use \({{\bf{p}}_2}\) as the origin of the coordinate system.]

Let \(W = \left\{ {{{\bf{v}}_1},......,{{\bf{v}}_p}} \right\}\). Show that if \({\bf{x}}\) is orthogonal to each \({{\bf{v}}_j}\), for \(1 \le j \le p\), then \({\bf{x}}\) is orthogonal to every vector in \(W\).
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