3. Use the definition of convex sets to answer the following: (a) Show that if the sets S and T are convex, then SnT is convex. (b) Show that the intersection of any number of convex sets is convex. (c) A hyperplane in Rd is a set of points of the form {r: ax=b} where a R and b E R. Show that hyperplanes are convex. Hint: If you're having trouble seeing why this is true in general, try the problem with a simple concrete example in 2-dimensions.
3. Use the definition of convex sets to answer the following: (a) Show that if the sets S and T are convex, then SnT is convex. (b) Show that the intersection of any number of convex sets is convex. (c) A hyperplane in Rd is a set of points of the form {r: ax=b} where a R and b E R. Show that hyperplanes are convex. Hint: If you're having trouble seeing why this is true in general, try the problem with a simple concrete example in 2-dimensions.
Elementary Geometry For College Students, 7e
7th Edition
ISBN:9781337614085
Author:Alexander, Daniel C.; Koeberlein, Geralyn M.
Publisher:Alexander, Daniel C.; Koeberlein, Geralyn M.
Chapter5: Similar Triangles
Section5.2: Similar Polygons
Problem 11E: a Does the similarity relationship have a reflexive property for triangles and polygons in general?...
Related questions
Question
Q3
![3. Use the definition of convex sets to answer the following:
(a) Show that if the sets S and T are convex, then SnT is convex.
(b) Show that the intersection of any number of convex sets is convex.
(c) A hyperplane in Rd is a set of points of the form {r: ax=b} where a € Rd
and bER. Show that hyperplanes are convex. Hint: If you're having trouble
seeing why this is true in general, try the problem with a simple concrete
example in 2-dimensions.
(d) A halfspace R is a set of points of the form {r: a'r <b} where a € Rd and
bER. Show that halfspaces are convex.
(e) Using (a) and (d), show that {r:c<a¹a<b} is convex when c < b.
(f) Using (b) and (e), show that the d-dimensional cube
{a: 0≤x≤1 for i {1,..., d}}
is convex](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F66f2d714-57ab-40fc-941f-d13524a78f49%2F5b460207-6ffc-41c6-8ae1-0321b2fb02ec%2Fpmi68e9_processed.png&w=3840&q=75)
Transcribed Image Text:3. Use the definition of convex sets to answer the following:
(a) Show that if the sets S and T are convex, then SnT is convex.
(b) Show that the intersection of any number of convex sets is convex.
(c) A hyperplane in Rd is a set of points of the form {r: ax=b} where a € Rd
and bER. Show that hyperplanes are convex. Hint: If you're having trouble
seeing why this is true in general, try the problem with a simple concrete
example in 2-dimensions.
(d) A halfspace R is a set of points of the form {r: a'r <b} where a € Rd and
bER. Show that halfspaces are convex.
(e) Using (a) and (d), show that {r:c<a¹a<b} is convex when c < b.
(f) Using (b) and (e), show that the d-dimensional cube
{a: 0≤x≤1 for i {1,..., d}}
is convex
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 3 steps with 2 images
![Blurred answer](/static/compass_v2/solution-images/blurred-answer.jpg)
Recommended textbooks for you
![Elementary Geometry For College Students, 7e](https://www.bartleby.com/isbn_cover_images/9781337614085/9781337614085_smallCoverImage.jpg)
Elementary Geometry For College Students, 7e
Geometry
ISBN:
9781337614085
Author:
Alexander, Daniel C.; Koeberlein, Geralyn M.
Publisher:
Cengage,
![Calculus For The Life Sciences](https://www.bartleby.com/isbn_cover_images/9780321964038/9780321964038_smallCoverImage.gif)
Calculus For The Life Sciences
Calculus
ISBN:
9780321964038
Author:
GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Publisher:
Pearson Addison Wesley,
![Elementary Geometry for College Students](https://www.bartleby.com/isbn_cover_images/9781285195698/9781285195698_smallCoverImage.gif)
Elementary Geometry for College Students
Geometry
ISBN:
9781285195698
Author:
Daniel C. Alexander, Geralyn M. Koeberlein
Publisher:
Cengage Learning
![Elementary Geometry For College Students, 7e](https://www.bartleby.com/isbn_cover_images/9781337614085/9781337614085_smallCoverImage.jpg)
Elementary Geometry For College Students, 7e
Geometry
ISBN:
9781337614085
Author:
Alexander, Daniel C.; Koeberlein, Geralyn M.
Publisher:
Cengage,
![Calculus For The Life Sciences](https://www.bartleby.com/isbn_cover_images/9780321964038/9780321964038_smallCoverImage.gif)
Calculus For The Life Sciences
Calculus
ISBN:
9780321964038
Author:
GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Publisher:
Pearson Addison Wesley,
![Elementary Geometry for College Students](https://www.bartleby.com/isbn_cover_images/9781285195698/9781285195698_smallCoverImage.gif)
Elementary Geometry for College Students
Geometry
ISBN:
9781285195698
Author:
Daniel C. Alexander, Geralyn M. Koeberlein
Publisher:
Cengage Learning
Algebra & Trigonometry with Analytic Geometry
Algebra
ISBN:
9781133382119
Author:
Swokowski
Publisher:
Cengage
![College Algebra](https://www.bartleby.com/isbn_cover_images/9781337282291/9781337282291_smallCoverImage.gif)
![Trigonometry (MindTap Course List)](https://www.bartleby.com/isbn_cover_images/9781337278461/9781337278461_smallCoverImage.gif)
Trigonometry (MindTap Course List)
Trigonometry
ISBN:
9781337278461
Author:
Ron Larson
Publisher:
Cengage Learning