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Chapter 8: Q16E (page 437)

Question: In Exercise 16 and 17, mark each statement True or False. Justify each answer.

a. A polytope is the convex hull of a finite set of points.

b. Let p be an extreme point of convex set S. If \({\bf{u}},\,{\bf{v}}\, \in S\), \({\bf{p}} \in \overline {{\bf{uv}}} \), and \({\bf{p}} \ne {\bf{u}}\), then \({\bf{p}} = {\bf{v}}\).

c. If S is a nonempty convex subset of \({\mathbb{R}^n}\), then S is the convex hull of its profile.

d. The 4-dimensional simplex \({S^{\bf{4}}}\) have exactly five facets, each of which is a 3-dimensional tetrahedron.

Short Answer

Expert verified

a. The given statement is True.

b. The given statement is True.

c. The given statement is False.

d. The given statement is True.

Step by step solution

01

Check for the statement (a)

A polytope in \({\mathbb{R}^n}\) is a convex hullwith finite set of points. If P is a polytope of dimension k then P is called a k polytope.

So, the given statement (a) is True.

02

Check for statement (b)

A point p in a vertex set S is called an extreme point of S if p is not the interior of any line segment in set S.

So, the given statement (b) is True.

03

Check for statement (c)

From theorem-16,If S is a nonempty convex subset of\({\mathbb{R}^n}\), thenS must be compact.

So, the given statement is False.

04

Check for statement (d)

The fifth vertex of \({s^4}\) lies inside \({s^3}\).

A 4-dimensional simplexhas five facets, and all of them are 3-dimensional tetrahedrons.

So, the given statement is True.

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Most popular questions from this chapter

In Exercises 7 and 8, find the barycentric coordinates of p with respect to the affinely independent set of points that precedes it.

7. \(\left( {\begin{array}{{}}1\\{ - 1}\\2\\1\end{array}} \right),\left( {\begin{array}{{}}2\\1\\0\\1\end{array}} \right),\left( {\begin{array}{{}}1\\2\\{ - 2}\\0\end{array}} \right)\), \({\mathop{\rm p}\nolimits} = \left( {\begin{array}{{}}5\\4\\{ - 2}\\2\end{array}} \right)\)

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14.a. Justify each equal sign:

\(3\left( {{{\mathop{\rm r}\nolimits} _3} - {{\mathop{\rm r}\nolimits} _2}} \right) = z'\left( 1 \right) = .5x'\left( 1 \right) = \frac{3}{2}\left( {{{\mathop{\rm p}\nolimits} _3} - {{\mathop{\rm p}\nolimits} _2}} \right)\)

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d. Use part (c) to show that \(8{{\mathop{\rm r}\nolimits} _1} = - {{\mathop{\rm p}\nolimits} _0} - {{\mathop{\rm p}\nolimits} _1} + {{\mathop{\rm p}\nolimits} _2} + {{\mathop{\rm p}\nolimits} _3} + 8{{\mathop{\rm r}\nolimits} _0}\).

e. Use part (d) equation (8), and part (a) to show that \({{\mathop{\rm r}\nolimits} _1}\) is the midpoint of the segment from \({{\mathop{\rm r}\nolimits} _2}\) to the midpoint of the segment from \({{\mathop{\rm p}\nolimits} _1}\) to \({{\mathop{\rm p}\nolimits} _2}\). That is, \({{\mathop{\rm r}\nolimits} _1} = \frac{1}{2}\left( {{{\mathop{\rm r}\nolimits} _2} + \frac{1}{2}\left( {{{\mathop{\rm p}\nolimits} _1} + {{\mathop{\rm p}\nolimits} _2}} \right)} \right)\).

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\({\bf{x}}\left( t \right) = G{M_S}{\bf{u}}\left( t \right)\)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 \({\bf{x}}\left( t \right)\).

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5.\(\left( {\begin{aligned}{{}}1\\0\\{ - 2}\end{aligned}} \right),\left( {\begin{aligned}{{}}0\\1\\1\end{aligned}} \right),\left( {\begin{aligned}{{}}{ - 1}\\5\\1\end{aligned}} \right),\left( {\begin{aligned}{{}}0\\5\\{ - 3}\end{aligned}} \right)\)
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