- Some Geometry With Cones Problem Suppose that C is a cone in R" and let us define another set C' in R" as C = {ye R"ly¹x ≤ 0 for all x = C}. Prove that C' is also a cone in R". Determine the intersection between C and C'. Construct C' from C if C is the polyhedron cone XI C = - {[ + ]« R²|1 - 292 50,-25 + x2 50}. ≤0,-2x1 X2 and plot both C and C' in the R² plane.
- Some Geometry With Cones Problem Suppose that C is a cone in R" and let us define another set C' in R" as C = {ye R"ly¹x ≤ 0 for all x = C}. Prove that C' is also a cone in R". Determine the intersection between C and C'. Construct C' from C if C is the polyhedron cone XI C = - {[ + ]« R²|1 - 292 50,-25 + x2 50}. ≤0,-2x1 X2 and plot both C and C' in the R² plane.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![- Some Geometry With Cones
Problem
Suppose that C is a cone in R" and let us define another set C' in R" as
C' = {y = R"y¹x ≤ 0 for all x € C}.
Prove that C' is also a cone in R".
Determine the intersection between C and C'.
Construct C' from Cif C is the polyhedron cone
XI
C =
- { [ ] = 4²| 41-2502 50,-241 + x2 50}.
≤0,-2x₁
X2
and plot both C and C' in the R² plane.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb8a9802b-ded9-4a7b-93c8-71218002814a%2F54f97a92-44e8-4f06-8624-a87997055a1c%2Fwi4r0gm_processed.png&w=3840&q=75)
Transcribed Image Text:- Some Geometry With Cones
Problem
Suppose that C is a cone in R" and let us define another set C' in R" as
C' = {y = R"y¹x ≤ 0 for all x € C}.
Prove that C' is also a cone in R".
Determine the intersection between C and C'.
Construct C' from Cif C is the polyhedron cone
XI
C =
- { [ ] = 4²| 41-2502 50,-241 + x2 50}.
≤0,-2x₁
X2
and plot both C and C' in the R² plane.
![Cones
A subset set C≤R" is a called a cone in R" if, for all x in C, 2x is also in C for all λ > 0. It
should be noted that as long that C is not empty, the origin x = 0 must be in the cone. To see
this, suppose that C is not empty so that some point x is in C, then 2x = C for all > > 0 and
setting λ = 0, we find that Ox C or simply 0 C. Thus we find that
C + Ø
0 € C.
Of course 0 € C ⇒ C + Ø and so we see that
C + Ø
0 € C.
A cone that is also a convex set is called a convex cone and we say that a convex cone is
pointed if it contains the origin 0 as an extreme point.
A Polyhedral Cone
A polyhedron of the form
P = {x € R" | Ax ≥ 0}
is a cone known as a polyhedral cone because if x = P, so that Ax ≥ 0, then (for all λ ≥ 0),
2Ax ≥ 20, or A(2x) ≥ 0, which says that λx € P. We note that P is not empty since
x = 0 € P. Note that
X1
X1
P =
- {*]*[
[ * ] }
X2
X2
is a polyhedron cone that is not pointed, and is shown in the figure below.
ER² 1 1
P =
P is everything above the line
x₁ + x₂ = 0 and is not pointed
Suppose also that A is a full rank m x n matrix with m≥n. Then since all the m constraints
Ax ≥ 0 are active at x = 0 € R" and m≥n, we see that x = 0 will be an extreme point of P
showing then that P is pointed for m≥n. For example, the cone
1
-{][I][:]
{*
R²
X2
21
is a polyhedron cone that is pointed, and is shown in the figure below.
XI
X2
P is everything above the lines x₁ + x₂ = 0
and 2x1 + x₂ = 0 and is pointed
0
0
Theorem #10: Cones and Convex Sets
A cone is a convex set if and only if it is closed under vector addition. Recall that a set in R¹
is closed under vector addition if for any xe C and y e C, x+y e C.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb8a9802b-ded9-4a7b-93c8-71218002814a%2F54f97a92-44e8-4f06-8624-a87997055a1c%2F8hwln_processed.png&w=3840&q=75)
Transcribed Image Text:Cones
A subset set C≤R" is a called a cone in R" if, for all x in C, 2x is also in C for all λ > 0. It
should be noted that as long that C is not empty, the origin x = 0 must be in the cone. To see
this, suppose that C is not empty so that some point x is in C, then 2x = C for all > > 0 and
setting λ = 0, we find that Ox C or simply 0 C. Thus we find that
C + Ø
0 € C.
Of course 0 € C ⇒ C + Ø and so we see that
C + Ø
0 € C.
A cone that is also a convex set is called a convex cone and we say that a convex cone is
pointed if it contains the origin 0 as an extreme point.
A Polyhedral Cone
A polyhedron of the form
P = {x € R" | Ax ≥ 0}
is a cone known as a polyhedral cone because if x = P, so that Ax ≥ 0, then (for all λ ≥ 0),
2Ax ≥ 20, or A(2x) ≥ 0, which says that λx € P. We note that P is not empty since
x = 0 € P. Note that
X1
X1
P =
- {*]*[
[ * ] }
X2
X2
is a polyhedron cone that is not pointed, and is shown in the figure below.
ER² 1 1
P =
P is everything above the line
x₁ + x₂ = 0 and is not pointed
Suppose also that A is a full rank m x n matrix with m≥n. Then since all the m constraints
Ax ≥ 0 are active at x = 0 € R" and m≥n, we see that x = 0 will be an extreme point of P
showing then that P is pointed for m≥n. For example, the cone
1
-{][I][:]
{*
R²
X2
21
is a polyhedron cone that is pointed, and is shown in the figure below.
XI
X2
P is everything above the lines x₁ + x₂ = 0
and 2x1 + x₂ = 0 and is pointed
0
0
Theorem #10: Cones and Convex Sets
A cone is a convex set if and only if it is closed under vector addition. Recall that a set in R¹
is closed under vector addition if for any xe C and y e C, x+y e C.
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