iven a space
we can separate the last variable
and call a hyperplane vertical if its normal vector is of the form
.
A set
for a fixed
is called a vertical line.

Proposition

(Nonvertical separation). Let
be a nonempty convex subset of
that
contains no vertical lines. Then:

1. The
is contained in a half-space of a nonvertical hyperplane.

2. If
then there is a nonvertical hyperplane that separates
and
.

Proof

1. By contradiction and proposition
(
Intersection of half-spaces
), if
all half-spaces that surround
come from vertical hyperplanes then
must have a vertical line.

2. Consider
.
If
is not of the form
then we are done. If its is of the form
then we use a perturbation on the figure
(
Nonvertical separation figure
).
First, take any hyperplane from the part (1) of the statement. There are no
points of
in the part of the space below the broken plane (A,O,D). We perform a slight
perturbation
the hyperplane (A,B) into that area while maintaining separation from the
point
.