*n*-var LP's require an *n*-dimensional space to ``geometrically''
represent a vector corresponding to a pricing of their decision
variables. However, the concepts and techniques that allowed the
geometric representation of the 2-var LP's in the previous section,
generalize quite straightforwardly in this more complicated case.

Hence, given a linear constraint:

and a point satisfying Equation 14 as equality, we can perceive the solution space of

- Equation as the set of points for which the vector is at right angles with vector ,
- Inequality as the set of points for which the vector forms an acute angle with vector , and
- Inequality as the set of points for which the vector forms an obtuse angle with vector .

From the above description, it follows that the solution space of the
equation , i.e., the 3-dim case, is a
*plane* perpendicular to vector , which is
characterized as the plane *normal*. Since this normality
concept carries over to the more general *n*-dim case, we characterize
the solution space of an *n*-var linear equation as a *
hyperplane*. Furthermore, similar to the 2-var LP case, a hyperplane
defined by the equation ,
divides the *n*-dim space into two *half-spaces*: one of them is
the solution space of the inequality , and the other one is the solution space of the inequality
.

Hence, the solution space (*feasible region*) of an *n*-var LP is
geometrically defined by the intersection of a number of half-spaces
and/or hyperplanes equal to the LP constraints, including the *
sign restrictions*. Such a set is characterized as a *
polytope*. In particular, a bounded polytope is called a *
polyhedron*.

Two other concepts of interest in the following discussion are those
of the *straight line* and the *line segment*. Given two
points and , the *straight line*
passing through them is algebraicaly defined by the set of
points

Figure 7 provides a visualization of this definition.

**Figure 7:** The equation of a straight line

As it is seen in the figure above, the *line segment* between
points and is analytically defined by:

Notice that Equation 16 can be rewritten as:

or

We say that Equations 17 and 18
define a *convex combination* of points and .

Fri Jun 20 15:03:05 CDT 1997