Mixing
Decomposing a polyhedron in $mathbb{R}^3$ into a lineality space, cone, and polytope.
$begingroup$
Consider the set in $mathbb{R}^3$ given by $${(x,y,z) : x+y+zge 3, x ge 0}$$
I can picture this set; it is simply the intersection of two half-spaces. A theorem states that every polyhedron can be decomposed as the direct sum of a lineality space, a cone, and a polytope. I am attempting to figure out what those are for this problem.
This set can be constructed as a system of equations $Amathbf{x} le mathbf{b}$, where $$A = begin{pmatrix}-1&-1&-1\-1&0&0
end{pmatrix},quadmathbf{x} = begin{pmatrix} x\y\zend{pmatrix}, quad text{ and }quad mathbf{b} = begin{pmatrix} -3\0end{pmatrix}$$
The lineality space is defined to be $N(A)$.
The cone is defined to be ${mathbf{x} : Amathbf{x} le mathbf{0}}$
The polytope is a bounded polyhedron.
I have concluded that the lineality space is span${ (0,-1,1)}$ (a line).
I am less confident on my classification of the cone and polytope though. I hypothesize that my cone is all the conic combinations of the vectors ${(0,1,1), (2,-1,-1), (0,-1,1), (0,1,-1)}$. I also hypothesize that my polytope is just the origin.
If anyone can help me to either verify my solutions or show me why my hypotheses are incorrect, I'd be very grateful. Thanks in advance.
linear-programming polyhedra polytopes convex-cone
asked Jan 31 at 1:15
BSplitterBSplitter
572215
$endgroup$
add a comment |
$begingroup$
Consider the set in $mathbb{R}^3$ given by $${(x,y,z) : x+y+zge 3, x ge 0}$$
I can picture this set; it is simply the intersection of two half-spaces. A theorem states that every polyhedron can be decomposed as the direct sum of a lineality space, a cone, and a polytope. I am attempting to figure out what those are for this problem.
This set can be constructed as a system of equations $Amathbf{x} le mathbf{b}$, where $$A = begin{pmatrix}-1&-1&-1\-1&0&0
end{pmatrix},quadmathbf{x} = begin{pmatrix} x\y\zend{pmatrix}, quad text{ and }quad mathbf{b} = begin{pmatrix} -3\0end{pmatrix}$$
The lineality space is defined to be $N(A)$.
The cone is defined to be ${mathbf{x} : Amathbf{x} le mathbf{0}}$
The polytope is a bounded polyhedron.
I have concluded that the lineality space is span${ (0,-1,1)}$ (a line).
I am less confident on my classification of the cone and polytope though. I hypothesize that my cone is all the conic combinations of the vectors ${(0,1,1), (2,-1,-1), (0,-1,1), (0,1,-1)}$. I also hypothesize that my polytope is just the origin.
If anyone can help me to either verify my solutions or show me why my hypotheses are incorrect, I'd be very grateful. Thanks in advance.
linear-programming polyhedra polytopes convex-cone
asked Jan 31 at 1:15
BSplitterBSplitter
572215
$endgroup$
add a comment |
$begingroup$
Consider the set in $mathbb{R}^3$ given by $${(x,y,z) : x+y+zge 3, x ge 0}$$
I can picture this set; it is simply the intersection of two half-spaces. A theorem states that every polyhedron can be decomposed as the direct sum of a lineality space, a cone, and a polytope. I am attempting to figure out what those are for this problem.
This set can be constructed as a system of equations $Amathbf{x} le mathbf{b}$, where $$A = begin{pmatrix}-1&-1&-1\-1&0&0
end{pmatrix},quadmathbf{x} = begin{pmatrix} x\y\zend{pmatrix}, quad text{ and }quad mathbf{b} = begin{pmatrix} -3\0end{pmatrix}$$
The lineality space is defined to be $N(A)$.
The cone is defined to be ${mathbf{x} : Amathbf{x} le mathbf{0}}$
The polytope is a bounded polyhedron.
I have concluded that the lineality space is span${ (0,-1,1)}$ (a line).
I am less confident on my classification of the cone and polytope though. I hypothesize that my cone is all the conic combinations of the vectors ${(0,1,1), (2,-1,-1), (0,-1,1), (0,1,-1)}$. I also hypothesize that my polytope is just the origin.
If anyone can help me to either verify my solutions or show me why my hypotheses are incorrect, I'd be very grateful. Thanks in advance.
linear-programming polyhedra polytopes convex-cone
asked Jan 31 at 1:15
BSplitterBSplitter
572215
$endgroup$
Consider the set in $mathbb{R}^3$ given by $${(x,y,z) : x+y+zge 3, x ge 0}$$
I can picture this set; it is simply the intersection of two half-spaces. A theorem states that every polyhedron can be decomposed as the direct sum of a lineality space, a cone, and a polytope. I am attempting to figure out what those are for this problem.
This set can be constructed as a system of equations $Amathbf{x} le mathbf{b}$, where $$A = begin{pmatrix}-1&-1&-1\-1&0&0
end{pmatrix},quadmathbf{x} = begin{pmatrix} x\y\zend{pmatrix}, quad text{ and }quad mathbf{b} = begin{pmatrix} -3\0end{pmatrix}$$
The lineality space is defined to be $N(A)$.
The cone is defined to be ${mathbf{x} : Amathbf{x} le mathbf{0}}$
The polytope is a bounded polyhedron.
I have concluded that the lineality space is span${ (0,-1,1)}$ (a line).
I am less confident on my classification of the cone and polytope though. I hypothesize that my cone is all the conic combinations of the vectors ${(0,1,1), (2,-1,-1), (0,-1,1), (0,1,-1)}$. I also hypothesize that my polytope is just the origin.
If anyone can help me to either verify my solutions or show me why my hypotheses are incorrect, I'd be very grateful. Thanks in advance.
linear-programming polyhedra polytopes convex-cone
linear-programming polyhedra polytopes convex-cone
asked Jan 31 at 1:15
BSplitterBSplitter
572215
asked Jan 31 at 1:15
BSplitterBSplitter
572215
asked Jan 31 at 1:15
BSplitterBSplitter
572215
asked Jan 31 at 1:15
BSplitterBSplitter
572215
asked Jan 31 at 1:15
BSplitterBSplitter
572215
572215
add a comment |
add a comment |
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