| Popis: |
In this paper we characterize the set of orthogonal polyhedra in R 3 that minimize the sum of their internal solid angles. To prove our result, we first generalize the well known result that any orthogonal polygon in the plane with n vertices has ( n + 4 ) / 2 convex and ( n − 4 ) / 2 reflex vertices; see [1] , [2] . We prove that an orthogonal polyhedron of genus g with n vertices, k of which have degree greater than or equal to 4, has ( n + 8 − 8 g + 3 k ) / 2 convex vertices and ( n − 8 + 8 g − 3 k ) / 2 reflex vertices. We also prove that the sum of the solid angles of an orthogonal polyhedron is at least ( n − 4 + 4 g ) π , and at most ( 3 n − 24 − 4 g ) π . These inequalities are minimized and maximized, respectively, when g = 0 . |
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