et
be a subset of
.
The
may have common points with the
th
coordinate axis. We introduce the
quantity
A normal vector to a nonvertical hyperplane may be normalized to a form
.
A nonvertical hyperplane that crosses the
th
coordinate axis at the point
and has a normal vector
has the
representation
Indeed,
The set
is contained in the upper half plane of
iff
Hence, the
quantity
is the maximum
th
axis crossing level for all hyperplanes that contain the set
in the upper half space and have the normal vector
.
The
is a concave function.
We introduce the
quantity
Proof
.
We investigate the conditions for the equality
Observe that by definition of these quantities all that is needed is existence
of a supporting hyperplane at a point
.
The pictures (
Crossing points figure
1
)(
Crossing points figure 3
)
show basic examples when this may or may not happen.
Crossing points figure 2. The
upper boundary is included in the set. The other boundaries are
excluded.

Crossing theorem 2
figure

