(Convex set). The set
is called "the line segment between
and
".
The set
is called "convex" iff
for
.
Dimension of the convex set is the dimension of its affine hull.
Intersection of convex sets is a convex set. Consequently, for any collections
of numbers
and points
the set
is convex.
Definition
(Convex hull). The convex hull
of any set
is the intersection of all convex sets that contain
.
If the collection of numbers
is such that
and
then the sum
is called "the convex combination of points
".
Proposition
The convex hull of set S consists of all convex combinations of all elements
of S.
Definition
(Convex Hull Cone Relative
Interior). The set
is called a "cone" if it is closed with respect to positive scalar
multiplication:
for
and
.
The convex cone "generated by the set
and denoted
"
"
is the convex hull of all the lines joining all points of
with the origin. Let
denote a unit ball. The "closure" of the set
is the set
.
The "relative interior" is the set
.
