Principles of Mathematics in Operations Research

Solutions 243

Pn+\ is not a union of a cone at XQ and a poly tope.

Pn+i is a direct sum of a cone at xo and Cn.

P„+i is an intersection of a cone at x 0 and Cn+i provided that XQ £ Cn+i\Cn.

7.5 See Figure S.9.

(0,1,1

'• •

(1,1,0)

^r ,

%!

(1,0,1)

Fig. S.9. A tetrahedron

The diagonal ray (1,1,1)T of the cube is orthogonal to facet F4. Thus, F4 =

{x e R^3 : xi + X2+ x 3 = a}. Since this facet contains (0,1,1)T, (1,0, l)r,

(1,1,0)T, the value of a is 2. Therefore,

F 4 = {x 6 K^3 : xi + x 2 + x 3 = 2}.

Since (0,0,0)T is on the tetrahedron, the following halfspace is valid and facet
defining
ff 4 ={i6i^3 : xi+x 2 + x 3 < 2}
Similarly,
Fi = {x £ R^3 : xi - x 2 - x 3 = 0} ,

F 2 = {x e R^3 : -X! + x 2 - x 3 = 0} ,

F 3 = {ieR^3 : -xi - x 2 + x 3 = 0}.
The following set describes the tetrahedron:

xi +x 2 + x 3 < 2,

xi - x 2 - x 3 < 0,

—x\ + x 2 - x 3 < 0,

—xi — x 2 + x 3 < 0.