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?...
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Q3
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
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