Principles of Mathematics in Operations Research

Solutions 241

Problems of Chapter 7

7.1
a) A zero dimensional polytope is a point.
b) One dimensional polytopes are line segments.
c) Two dimensional polytopes are n-gons:
triangle (3), rectangle (4), trapezoid (4), pentagon (5),

7.2 Zi 2 =conv(ei,e 2 ,e3). See Figure S.6.

+1- /

(1,0,0) xi

Fig. S.6. A 2 in

7.3 C 3 =conv((0,0,0)T, (a, 0,0)T, (0, a, 0)T, (0,0, a)T, (a, a, 0)T,
(a, 0, a)T, (0, a, a)T, (a, a, a)T)

Cn = {xeWl : 0 <Xi < a, i = l,...,n; a €R+}.

A k

-4-

f ~

CUBE OCTAHEDRON

Fig. S.7. Cube and octahedron

C 34 =conv((a, 0,0)T, (0, a, 0)T, (0,0, a)T, (-a, 0,0)T, (0, -a, 0)T, (0,0, -a)T)

C* = J x e Kn : ^ |xi| <a,aER+.