/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Q15E A k-pyramid \({P^k}\) is the con... [FREE SOLUTION] | 91影视

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Chapter 8: Q15E (page 437)

A k-pyramid \({P^k}\) is the convex hull of a \(\left( {k - {\bf{1}}} \right)\)-polytope Q and a point \({\bf{x}} \notin {\bf{aff}}\,\,Q\). Find a formula for each of the following in terms of \({f_j}\left( Q \right),j = {\bf{0}},......,n - {\bf{1}}\).

a. The number of vertices of \({P^n}\): \({f_{\bf{0}}}\left( {{P^n}} \right)\).

b. The number of k-faces of \({P^n}\): \({f_k}\left( {{P^n}} \right)\), for \({\bf{1}} \le k \le n - {\bf{2}}\)

c. The number of \(\left( {n - {\bf{1}}} \right)\) dimensional facets of \({P^n}\): \({f_{n - {\bf{1}}}}\left( {{P^n}} \right)\).

Short Answer

Expert verified

a. \({f_0}\left( {{P^n}} \right) = {f_0}\left( Q \right) + 1\)

b. \({f_k}\left( {{P^n}} \right) = {f_k}\left( Q \right) + {f_{k - 1}}\left( Q \right)\)

c. \({f_{n - 1}}\left( {{P^n}} \right) = {f_{n - 2}}\left( Q \right) + 1\)

Step by step solution

01

Find the formula for part (a)

As k pyramid is the convex hull of \(\left( {k - 1} \right)\) polytope Q and \(x \notin {\rm{aff}}\,\,Q\), then the formula for vertices of \({P_n}\) is shown below:

\({f_0}\left( {{P^n}} \right) = {f_0}\left( Q \right) + 1\)

Thus, the formula is \({f_0}\left( {{P^n}} \right) = {f_0}\left( Q \right) + 1\).

02

Find the formula for part (b)

The number of k-faces of \({P_n}\) is given by the formula as shown below:

\({f_k}\left( {{P^n}} \right) = {f_k}\left( Q \right) + {f_{k - 1}}\left( Q \right)\)

Thus, the formula is \({f_k}\left( {{P^n}} \right) = {f_k}\left( Q \right) + {f_{k - 1}}\left( Q \right)\).

03

Find the formula for part (c)

The number of \(n - 1\) dimensional faces of \({P^n}\) is given by the formula\({f_{n - 1}}\left( {{P^n}} \right) = {f_{n - 2}}\left( Q \right) + 1\).

Thus, the formula is \({f_{n - 1}}\left( {{P^n}} \right) = {f_{n - 2}}\left( Q \right) + 1\).

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Most popular questions from this chapter

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 1: Given points \({{\mathop{\rm p}\nolimits} _1} = \left( {\begin{array}{*{20}{c}}1\\0\end{array}} \right),{\rm{ }}{{\mathop{\rm p}\nolimits} _2} = \left( {\begin{array}{*{20}{c}}2\\3\end{array}} \right),\) and \({{\mathop{\rm p}\nolimits} _3} = \left( {\begin{array}{*{20}{c}}{ - 1}\\2\end{array}} \right)\) in \({\mathbb{R}^2}\), let \(S = {\mathop{\rm conv}\nolimits} \left\{ {{{\mathop{\rm p}\nolimits} _1},{{\mathop{\rm p}\nolimits} _2},{{\mathop{\rm p}\nolimits} _3}} \right\}\). For each linear functional \(f\), find the maximum value \(m\) of \(f\), find the maximum value \(m\) of \(f\) on the set \(S\), and find all points x in \(S\) at which \(f\left( {\mathop{\rm x}\nolimits} \right) = m\).

a.\(f\left( {{x_1},{x_2}} \right) = {x_1} - {x_2}\)

b. \(f\left( {{x_1},{x_2}} \right) = {x_1} + {x_2}\)

c. \(f\left( {{x_1},{x_2}} \right) = - 3{x_1} + {x_2}\)

Explain why a cubic Bezier curve is completely determined by \({\mathop{\rm x}\nolimits} \left( 0 \right)\), \(x'\left( 0 \right)\), \({\mathop{\rm x}\nolimits} \left( 1 \right)\), and \(x'\left( 1 \right)\).

Question: In Exercises 15-20, write a formula for a linear functional f and specify a number d, so that \(\left) {f:d} \right)\) the hyperplane H described in the exercise.

Let H be the plane in \({\mathbb{R}^{\bf{3}}}\) spanned by the rows of \(B = \left( {\begin{array}{*{20}{c}}{\bf{1}}&{\bf{4}}&{ - {\bf{5}}}\\{\bf{0}}&{ - {\bf{2}}}&{\bf{8}}\end{array}} \right)\). That is, \(H = {\bf{Row}}\,B\).

Question: 1. 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 \(\left( {\begin{array}{*{20}{c}}{\bf{3}}\\{\bf{1}}\end{array}} \right)\). Find a linear functional f and a real number d such that \(L = \left( {f:d} \right)\).
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