A hyperplane
may be represented as
for any fixed
.
Proposition
(Separating hyperplane theorem). If
the
are two nonempty disjoint convex sets then there is a hyperplane that
separates them.
Two nonempty convex disjoint sets
are not necessarily strictly separated. For example,
,
do not have a strictly separating hyperplane.
Let
and
be two disjoint closed convex subsets of
.
To investigate the conditions for
to be closed we introduce the subset
of
,
note that the transformation
is linear and seek to apply the proposition
(
Preservation of closeness
result
). We note that
is closed and
convex,
and
The condition
of the proposition (
Preservation
of closeness result
)
becomes
We arrive to the following additional sufficient conditions for strict
separation.
Proposition
(Intersection of halfspaces). The
closure of convex hull of a set
is the intersection of all closed halfspaces that contain
.
Proof
If there is a point in
that is not contained in the intersection of halfplanes then we arrive to
contradiction by using the theorem
(
Supporting hyperplane theorem
).
Let
be a line
for some
.
The definition of the proper separation requires that
is a single point or nothing and
consists of more then one point.
The sets
and
may not be properly separated.
The sets
and
are properly separated by the xaxis
.
