A small cover is a smooth closed manifold [M.sup.n] which admits a locally standard [Z.sup.n.sub.2]-action whose orbit space is a simple

convex polytope.

A

convex polytope satisfying these properties is called a Delzant polytope.

Consider a dimension n

convex polytope [DELTA] [subset] [([R.sup.n]).sup.*].

A DOP is a

convex polytope containing the object, constructed by taking a number k of appropriately oriented planes at infinity and bringing it closer to the object until they collide.

Any A(p(k)) and C(p(k)) belong to a

convex polytope [OMEGA] defined by

Note that any

convex polytope can be written in the form [[PI].sub.X](u) for suitable X and u.

The convexity theorem of Atiyah [1] and Guillemin-Sternberg [10] implies that [mu](M) is a

convex polytope in [R.sup.n].

[5] shows that every arrangement of spheres (and hence every central arrangemen of hyperplanes) is combinatorially equivalent to some

convex polytope, [9] proved that there is a relation between the number of lattice point on a sphere and the volume of it.

The point (Y1, Y2), governed by (18)-(19), can never leave the (N -1)-dimensional (here N = 3)

convex polytope and by definition [Y.sub.3] = 1 - [Y.sub.1] - [Y.sub.2].