Page 398 - A Course in Linear Algebra with Applications

382 Chapter Ten: Linear Programming

Convex subsets
n
Let Xi and X2 be two distinct points in R . The line
segment X\X2 joining X\ and X2 is defined to be the set of
points
{tX x + (1 - t)X 2 I 0 < t < 1}.
For example, if n < 3, the point tX\ + (1 — t)X2, where
0 < t < 1, is a typical point lying between Xi and X 2 on the
line which joins them. To see this one has to notice that

X 2 - (tX 1 + (1 - t)X 2) = t{X 2 - X x)

and
{tX x + (1 - t)X 2) - X, = (1 - i)(X 2 - Xi)

are parallel vectors.

(Keep in mind that we are using X to denote both the point
7
(xi,X2,xs) and the column vector {x\ X2 X3) '.)
A non-empty subset S of R n is called convex if, whenever
X\ and X 2 are points in S, every point on the line segment
is also a point of S.
X\X 2