Page 402 - A Course in Linear Algebra with Applications

386 Chapter Ten: Linear Programming

on m > 1, the claim being clearly true if m = 1. Now we have

m — 1
C
X
X = (1 - Cm) J2 iTZ—) i + mXm-
X Cm.
m—1
Next, since ^ c» = 1 — c m , we have
i-X
m—X
E l M _ C-i ' \ _ 1 _ r ^ m ~~ 1
*•
i = l - r ^
c
Also 0 < , i < 1 since Q < ci + • • • + c m _i = 1 — c m for
1 < i < m — 1. Hence
m—X

1
- - *^T77.
1 = 1
by the induction hypothesis on m. Finally,

X = (1 - c m )F + c mX m e T,

since T is convex.

Extreme points
n
Let S be a convex subset of R . A point of S is called an
extreme point if it is not an interior point of any line segment
joining two points of S. For example, the extreme points of
the set of points in the polygon below are just the six vertices
shown.