MATH5301 Elementary Analysis. Homework 11
11.1
Prove that the closure and the interior of a convex set A ⊂ R
n are also convex.
11.2
Prove that the intersection of an arbitrary collection of convex sets T
i∈I
Ci
is also convex.
11.3
Let {Ci}i∈N be a sequence of nested convex sets in R
n, i.e. Ci ⊂ Ci+1. Prove that S∞
i=1
Ci
is also convex.
11.4
(a) Show that the convex hull of any open set in R
n is open.
(b) Provide an example of a closed set A ⊂ R
n, such that its convex hull is not closed.
11.5
Let f : R
n → R be a convex function and A ⊂ R
n be a bounded set. Prove that f(A) is bounded in R.
11.6
Show that the convex hull of a compact set A ⊂ R
n is compact. (Hint: Caratheodory theorem)
