# 7D dissident maps - download pdf or read online By Ernst Dieterich and Lars Lindberg

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33 ei ai Fig. 34 We now construct the polyhedral angle V that is formed by the normals to the support planes of V . Consider the intersection of V with a plane R1 parallel to R. This intersection is the convex polygon Q1 whose vertices are the intersections of the edges of V with the plane R1 . It is easy to see that the extension of the perpendicular OO meets the plane R1 at some point O1 inside the polygon Q1 (since the plane perpendicular to OO at the point O is a support plane of V touching it only at the vertex O ).

An−1 (Fig. 27(a)). Let P stand for the convex hull of the ﬁgure formed by the given points A1 , . . , Am and the angle V . By assumption, P is a convex polyhedron with limit angle V . O O S Q a1 an an-1 Fig. 27(a) an Fig. 27(b) Translating the starting point of the half-line an , the nth edge of the angle V , to all points of the polyhedron P , we obtain a body P . 13 However, P is itself the convex hull of the ﬁgure formed by the points A1 , . . , Am and the 13 The polyhedron P is now unbounded; therefore, it is possible that a half-line going from the interior of P does not meet the surface of P .

13 However, P is itself the convex hull of the ﬁgure formed by the points A1 , . . , Am and the 13 The polyhedron P is now unbounded; therefore, it is possible that a half-line going from the interior of P does not meet the surface of P . However, this halfline then meets the surface if we extend it in the opposite direction. Otherwise P 36 1 Basic Concepts and Simplest Properties of Convex Polyhedra angle V , whereas V is (by Theorem 2) the convex hull of the set of its edges a1 , a2 , . . , an−1 .