Principles of Mathematics in Operations Research

100 7 Convex Sets

7.3. Cube and Octahedron

Characterize cubes and octahedrons with the help of three dimensional cube

C3, and octahedron C^.

7.4. Pyramid

Let Pn+i=conv(Cn,:Eo) be a (n+l)-dimensional pyramid, where XQ $ Cn.

Draw

P 3 = conv{C 2 : a = 1, (1/2,1/2,1)T)

and write down all describing inequalities.

7.5. Tetrahedron
The vertices of a tetrahedron of side length [2 can be given by a particularly
simple form when the vertices are taken as corners of the unit cube. Such a
tetrahedron inside a cube of side length 1 has side length \/2 with vertices
(0,0,0)T, (0,1,1)T, (1,0,1)T, (1,1,0)T. Draw and find a set of describing
inequalities. Is it possible to express Pn+\ as a union / intersection / direct
sum of a cone and a polytope?

7.6. Dodecahedron
Find the vertices of a dodecahedron (see Figure 7.6) of side length a = \/5 — 1.

Fig. 7.6. A dodecahedron

Web material

http://cepa.newschool.edu/het/essays/math/convex.htm

http://cm.bell-labs.eom/who/clarkson/cis677/lecture/6/index.html

http://cm.bell-labs.eom/who/clarkson/cis677/lecture/8/