1. Convex Sets 3
(1.3) Some interesting examples. Sometimes, it is very easy to see whether
the set is convex or not (see Figure 1).
convex convex non-convex
Figure 1. Two convex and one non-convex set
Sometimes, however, this is not so easy to see (cf. Problems 3, 4 and 5 below),
or a convex set may have a number of equivalent descriptions and their equivalence
may be not obvious (cf. Problems 6 and 7 below).
PROBLEMS.
We will encounter many of the harder problems later in the text.
1◦.
Prove that the convex hull of a set is a convex set.
2◦.
Let c1,... , cm be vectors from
Rd
and let β1,... , βm be numbers. The set
A = x
Rd
: ci,x βi for i = 1, . . . , m
is called a polyhedron. Prove that a polyhedron is a convex set.
3. Let v1,... , vm Rd be points. Let us fix positive numbers ρ1,... , ρm and
let us define a map: H : Rd −→ Rd by
H(x) =
1
f(x)
m
i=1
ρi exp x, vi vi, where f(x) =
m
i=1
ρi exp x, vi .
a◦)
Prove that the image of H lies in the convex hull of v1,... , vm.
b∗)
Prove that the image of H is convex and that for 0 and for any
y conv
(
v1,... , vm
)
there exists an x Rd such that dist
(any
H(x),y
)
.
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