iven a space
we can separate the last variable
and call a hyperplane vertical if its normal vector is of the form
for a fixed
is called a vertical line.
(Nonvertical separation). Let
be a nonempty convex subset of
contains no vertical lines. Then:
is contained in a half-space of a nonvertical hyperplane.
then there is a nonvertical hyperplane that separates
1. By contradiction and proposition
Intersection of half-spaces
all half-spaces that surround
come from vertical hyperplanes then
must have a vertical line.
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
in the part of the space below the broken plane (A,O,D). We perform a slight
the hyperplane (A,B) into that area while maintaining separation from the