2.11 Hyperbolic sets. Show that the hyperbolic set {r E R² | x1x2 > 1} is convex. As a generalization, show that {r E R | II"1xi > 1} is convex. Hint. If a, b > 0 and 0 <0<1, then aºbl- < 0a +(1 – 0)b; see §3.1.9. 2 12 Which of the follorin
2.11 Hyperbolic sets. Show that the hyperbolic set {r E R² | x1x2 > 1} is convex. As a generalization, show that {r E R | II"1xi > 1} is convex. Hint. If a, b > 0 and 0 <0<1, then aºbl- < 0a +(1 – 0)b; see §3.1.9. 2 12 Which of the follorin
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
Q 2.11 only handwritten solution accepted
![2.10 Solution set of a quadratic inequality. Let C C R" be the solution set of a quadratic
inequality,
C = {x € R" | a" Ax + b" x + c < 0},
with A E S", b E R", and c E R.
(a) Show that C is convex if A E 0.
(b) Show that the intersection of C and the hyperplane defined by g"x +h = 0 (where
g + 0) is convex if A + Agg" > 0 for some AE R.
Are the converses of these statements true?
2.11 Hyperbolic sets. Show that the hyperbolic set {x E R² | x1x2 > 1} is convex. As a
generalization, show that {x E R | II", xi > 1} is convex. Hint. If a, b > 0 and
0 <0< 1, then aºb'=° < 0a+ (1 – 0)b; see §3.1.9.
i=1
2.12 Which of the following sets are convex?
(a) A slab, i.e., a set of the form {x € R" | a < ax < }.
(b) A rectangle, i.e., a set of the form {x € R" | ai <xi < Bi, i = 1, ...,n}. A rectangle
is sometimes called a hyperrectangle when n > 2.
(c) A wedge, i.e., {r € R" | afx < b1, afa < b2}.
(d) The set of points closer to a given point than a given set, i.e.,
{x| ||x – xo||2 < |r – y||2 for all y E S}
where S C R".
(e) The set of points closer to one set than another, i.e.,
{x | dist(x, S) < dist(x,T)},
where S, T C R", and
dist(r, S) = inf{||r – 2||2 | 2 E S}.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F87ed42ca-73ec-45b5-916b-2849c1d41943%2Fccdd7fcf-d5df-4377-ab76-e4561f667eb7%2F9qb8s6d_processed.jpeg&w=3840&q=75)
Transcribed Image Text:2.10 Solution set of a quadratic inequality. Let C C R" be the solution set of a quadratic
inequality,
C = {x € R" | a" Ax + b" x + c < 0},
with A E S", b E R", and c E R.
(a) Show that C is convex if A E 0.
(b) Show that the intersection of C and the hyperplane defined by g"x +h = 0 (where
g + 0) is convex if A + Agg" > 0 for some AE R.
Are the converses of these statements true?
2.11 Hyperbolic sets. Show that the hyperbolic set {x E R² | x1x2 > 1} is convex. As a
generalization, show that {x E R | II", xi > 1} is convex. Hint. If a, b > 0 and
0 <0< 1, then aºb'=° < 0a+ (1 – 0)b; see §3.1.9.
i=1
2.12 Which of the following sets are convex?
(a) A slab, i.e., a set of the form {x € R" | a < ax < }.
(b) A rectangle, i.e., a set of the form {x € R" | ai <xi < Bi, i = 1, ...,n}. A rectangle
is sometimes called a hyperrectangle when n > 2.
(c) A wedge, i.e., {r € R" | afx < b1, afa < b2}.
(d) The set of points closer to a given point than a given set, i.e.,
{x| ||x – xo||2 < |r – y||2 for all y E S}
where S C R".
(e) The set of points closer to one set than another, i.e.,
{x | dist(x, S) < dist(x,T)},
where S, T C R", and
dist(r, S) = inf{||r – 2||2 | 2 E S}.
Expert Solution
![](/static/compass_v2/shared-icons/check-mark.png)
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
This is a popular solution!
Trending now
This is a popular solution!
Step by step
Solved in 2 steps with 2 images
![Blurred answer](/static/compass_v2/solution-images/blurred-answer.jpg)
Recommended textbooks for you
![Advanced Engineering Mathematics](https://www.bartleby.com/isbn_cover_images/9780470458365/9780470458365_smallCoverImage.gif)
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
![Numerical Methods for Engineers](https://www.bartleby.com/isbn_cover_images/9780073397924/9780073397924_smallCoverImage.gif)
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
![Introductory Mathematics for Engineering Applicat…](https://www.bartleby.com/isbn_cover_images/9781118141809/9781118141809_smallCoverImage.gif)
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
![Advanced Engineering Mathematics](https://www.bartleby.com/isbn_cover_images/9780470458365/9780470458365_smallCoverImage.gif)
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
![Numerical Methods for Engineers](https://www.bartleby.com/isbn_cover_images/9780073397924/9780073397924_smallCoverImage.gif)
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
![Introductory Mathematics for Engineering Applicat…](https://www.bartleby.com/isbn_cover_images/9781118141809/9781118141809_smallCoverImage.gif)
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
![Mathematics For Machine Technology](https://www.bartleby.com/isbn_cover_images/9781337798310/9781337798310_smallCoverImage.jpg)
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
![Basic Technical Mathematics](https://www.bartleby.com/isbn_cover_images/9780134437705/9780134437705_smallCoverImage.gif)
![Topology](https://www.bartleby.com/isbn_cover_images/9780134689517/9780134689517_smallCoverImage.gif)