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0 /. In fact the line segment used in the construction of ıP1 ;P2 2 D connects interior points in different facets Q1 ; Q2 of ˘ 0 and meets the interior of the face Q1 \ Q2 , dim Q1 \ Q2 D dim Q1 1 D dim Q2 1. ˘ 0 ; ˘ 00 /. g/ > 0 for all ı 0 2 D 0 . f / D 0 for all ı 2 D 0 . f C "g/ > 0 for all ı 2 D 00 . t u The definition of regular subdivision contains the following patching principle. 62. Let ˘ be a polytopal complex with regular subdivision ˘ 0 . Furthermore let ˘P00 be a regular subdivision of each facet P 2 ˘ 0 .

P [ P 0 /. As a last construction principle for polyhedra we introduce the join. Roughly speaking, it is the “free convex hull” that we obtain by considering polyhedra in positions independent of each other. Let P V and Q W be polyhedra. P [ Q/ will also be called the join of P and Q. 31 the join of P and Q is a polyhedron. P; Q/ is a polytope. Separation of polyhedra. A characteristic feature of convexity are separation theorems: convex sets that are disjoint or just touch each other can be separated by a hyperplane.

R. P / and there is no connected subset V P containing W strictly for which f jV extends to an affine mapping. If ˘ is a polyhedral complex and f W j˘ j ! R is a function, then a subset W of j˘ j is a domain of linearity of f if there exists a facet P 2 ˘ such that W P and W is a domain of linearity of f jP . 56. A subdivision ˘ 0 of a polyhedral complex ˘ is called regular if there is a convex function f W j˘ j ! R whose domains of linearity are the facets of ˘ 0 . Such a function f is called a support function for the subdivision ˘ 0 .