Cell Decomposition of Polytopes By Bending
Cell Decomposition of Polytopes By Bending
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. , P^ lying in its interior, each Pi separated by a hyperplane from the preceding ones, have been removed. We do not know whether the corresponding result holds without the separation condition.
4. Further generalizations.
Let r be a polytope containing in its interior, and let dV denote its boundary. Then V gives rise to a family C^, . . . , C^ of convex cones partitioning R** and to a convex function / : R*^ — > R whose restriction to each cone is a distinct linear function: let C* = coneo(P...t) for each facet Fi of F, and let / be the linear extension to each cone of the function defined by /(O) = 0, f{dT) = 1. It is well-known, however, that not every decomposition of R into convex cones with a common vertex arises from a polytope in this manner [9]; let us call one which does polytopal.
Theorem 1 then admits the following generalization: 7 Theorem 4. Let P be a d-polytope containing the origin O in its interior, and let C^ U • • ■ U C '', f : R'^ —^ R be a polytopal decomposition of R*^ such that each vertex of P lies in the interior of some C^ .
What to read after Cell Decomposition of Polytopes By Bending?
You can find similar books in the "Read Also" column, or choose other free books by Jacob Eli Goodman to read onlineMoreLess
To quickly assess the difficulty of the text, read a short excerpt:
. , P^ lying in its interior, each Pi separated by a hyperplane from the preceding ones, have been removed. We do not know whether the corresponding result holds without the separation condition.
4. Further generalizations.
Let r be a polytope containing in its interior, and let dV denote its boundary. Then V gives rise to a family C^, . . . , C^ of convex cones partitioning R** and to a convex function / : R*^ — > R whose restriction to each cone is a distinct linear function: let C* = coneo(P...t) for each facet Fi of F, and let / be the linear extension to each cone of the function defined by /(O) = 0, f{dT) = 1. It is well-known, however, that not every decomposition of R into convex cones with a common vertex arises from a polytope in this manner [9]; let us call one which does polytopal.
Theorem 1 then admits the following generalization: 7 Theorem 4. Let P be a d-polytope containing the origin O in its interior, and let C^ U • • ■ U C '', f : R'^ —^ R be a polytopal decomposition of R*^ such that each vertex of P lies in the interior of some C^ .
What to read after Cell Decomposition of Polytopes By Bending?
You can find similar books in the "Read Also" column, or choose other free books by Jacob Eli Goodman to read onlineMoreLess
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