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Unformatted text preview: b 1 u t 1 ut 2 c.
Since the numbers 1 u r 1 ur 2 and 1 u s 1 us 2 and 1 u t 1 ut 2 are all nonnegative and
since
1 u r 1 ur 2 1 u s 1 us 2 1 u t 1 ut 2 1 u r 1 s 1 t 1 u r 2 s 2 t 2
1 uu 1 we conclude that
1 u r1a s1b t1c u r2a s2b t2c H. 4. In an earlier exercise we saw the definition of the sum A B of two sets A and B of real numbers. A similar
definition can be given if A and B are subsets of R k :
a A and b B such that x a b
AB x
which we can write more briefly in the form
AB ab a A and b B. Prove that if A and B are convex subsets of R then the set A B is also convex.
Suppose that A and B are convex. Suppose that a 1 and a 2 belong to A and b 1 and b 2 belong to B,
and suppose that 0 t 1. We have
1 t a 1 b 1 t a 2 b 2 1 t a 1 ta 2 1 t b 1 tb 2
which is the sum of a member of A and a member of B.
k 5. If A is a subset of R k then a convex combination of members of A is defined to be any sum of the form
n r1a1 r2a2 rnan rjaj
j1 where n is a positive integer and each of the points a 1 , a 2 , , a n belongs to the set A and each of the
coefficients r 1 , r 2 , , r n is nonnegative and
n r j 1.
j1 Prove that if A is convex then every convex combination of members of A must belong to A. Solution: Suppose that A is a convex set. We prove the assertion by mathematical induction. For each integer n 2 we define p n to be the assertion that whenever r 1 , r 2 , , r n are nonnegative numbers
satisfying the condition
n rj 1
j1 and whenever a 1 , a 2 , , a n are points of the set A, we have
n rjaj A. j1 It follows at once from the convexity of the set A that the assertion p 2 is true. Now suppose that n is any
integer such that n 2 and such that the assertion p n is true. Suppose that r 1 , r 2 , , r n , r n1 are
nonnegative number, that
n 1 rj 1
j1 and that a 1 , a 2 , , a n , a n1 are all points of A. In the event that 110 n rj 0
j1 we have r n1 1 and
n r j a j a n 1 A. j1 We suppose now that
n rj 0 j1 and we define
n t rj.
j1 We observe that r n1 1 t and that
n 1 n rjaj r j a j r n 1 a n 1 j1 j1
n rj
a
tj t
j1 1 t a n 1 . Since the assertion p n is true we know that
n rj
a
tj A rjaj j1 A. and it follows from the convexity of A that
n 1
j1 6. Given a nonempty subset A of R k , the convex hull co A of A is defined to be the set of all possible convex
combinations of points of A.
a. Prove that if A is any nonempty subset of R k then A
This assertion is obvious. co A . b. Prove that if A and B are nonempty subsets of R k and A
This assertion is obvious. B then co A co B . c. Prove that if A is any nonempty subset of R k then co A is convex.
Suppose n is a positive integer, that a j and b j belong to A and that r j and s j are nonnegative
numbers for j 1, , n and suppose that
n n rj
j1 Suppose finally that 0 1. Then we have t n 1 s j 1.
j1 n n rjaj t t
j1 sjbj
j1 1 t r j a j ts j b j . j1 Since
n 1
j1 11...
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This note was uploaded on 11/26/2012 for the course MATH 2313 taught by Professor Staff during the Fall '08 term at Texas El Paso.
 Fall '08
 STAFF
 Math, Calculus

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