Polyhedral cone: Difference between revisions
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== Definition == |
== Definition == |
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Ziegler defines a polyhedral cone {{math|cone(''Y'')}} as the set of all linear combinations with nonnegative coefficients of a finite set of vectors {{mvar|Y}}.<ref>Ziegler, ''Lectures on Polytopes'' section 1.1</ref> In the case of the empty set, we define <math>\text{cone}(\emptyset) = \{\mathbf{0}\}</math>. |
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Equivalently, a polyhedral cone can be defined as the [[convex hull]] of the origin and a finite set of [[ray]]s extending from the origin. |
Equivalently, a polyhedral cone can be defined as the [[convex hull]] of the origin and a finite set of [[ray]]s extending from the origin. |
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Grunbaum<ref>Grunbaum, Branko. ''Convex Polytopes'', page 9.</ref> generalizes this definition to allow an apex other than the origin, notated <math>\text{cone}_a\, Y</math> where {{mvar|a}} is the apex. Otherwise, the definition is the same. |
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Revision as of 03:19, 2 September 2024
A polyhedral cone is a convex cone with a finite number of extreme rays.
Definition
Ziegler defines a polyhedral cone cone(Y) as the set of all linear combinations with nonnegative coefficients of a finite set of vectors Y .[1] In the case of the empty set, we define .
Equivalently, a polyhedral cone can be defined as the convex hull of the origin and a finite set of rays extending from the origin.
Grunbaum[2] generalizes this definition to allow an apex other than the origin, notated where a is the apex. Otherwise, the definition is the same.
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