By R. Schaback, K. Scherer
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24) is equivalent to the identity Ax D x for every x 2 . 1 fx0 g/ \ . 2 fx0 g/ and that the polyhedra A1 . 1 / C b 1 and A2 . n 1/-face if and only if this is true for the polyhedra A. 1 fx0 g/ and 2 fx0 g. A/ > 0. 25) do not hold, then we may replace 1 ; 1 2 fx0 g; 2 fx0 g, A, and I; respectively. n 1/-dimensional linear subspace L Â IRn . Hence there exists an orthogonal matrix Q such that QL D fx 2 IRn jxn D 0g. n 1/-face if and only if this property holds for the polyhedra Q 1 and Q 2 , and A1 x D x holds for every x 2 1 \ 2 if and only if QA1 QT y D y holds for every y 2 Q 1 \ Q 2 .
A set X Â IRn is called path connected if for any two points x0 ; x1 2 X there exists a continuous function p W Œ0; 1 ! 1/ D x1 . The function p is called a path joining x0 and x1 . Note that a path connected set is connected, but that the reverse statement fails in general. 1/. , for every path p W Œ0; 1 ! 1/ D x0 there exists a point x1 2 X and a continuous function H W Œ0; 1 Œ0; 1 ! 1; t/ for every t 2 Œ0; 1. Since X is assumed to be path connected, we may choose any point x1 2 X . With the aid of Browder’s Theorem and the Homotopy Lifting Theorem, one can prove the following result which relates local and global homeomorphisms.
I; i 0 ; j / with si;i 0 ;j . / D si;i 0 ;j . Q /, and since the interior of M. / is the solution set of the strict inequality system, we conclude that the face is proper, provided that intM. / ¤ ;. j If f is piecewise linear, then the we may choose ˛i D 0 for every j D 1; : : : ; m and every i D 1; : : : ; kj and thus the sets M. / are polyhedral cones. The next property of polyhedral subdivisions is a consequence of the decomposition theorem for polyhedra. 4. If ˙ is a polyhedral subdivision of a polyhedron P Â IRn , then all polyhedra 2 ˙ have the same lineality space.