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Unformatted text preview: 1 t r j ts j 1 the expression n
j1 t r j a j ts j b j is also a convex combination of points of A. 1 d. Prove that if A is a nonempty subset of R k then A is convex if and only if A co A .
The result follows at once from part c and Exercise 5.
e. Prove that if A and B are nonempty subsets of R k and A
The result follows at once from parts b and d.
7. B and B is convex then co A B. a. Prove that if A and B are nonempty subsets of R k then
co A B
co A co B .
We shall prove in part c that this inequality can be replaced by an equation.
Suppose that n is a positive integer, that a j A and b j B and r j 0 for j 1, , n and that
n
r 1. Then
j1 j
n n n rj aj bj
j1 rjaj
j1 rjbj co A co B . j1 b. Prove that if A and B are nonempty subsets of R k and x co A and y B then x y co A B .
We assume that x co A and that y B.
Choose a positive integer n and members a j of the set A and numbers r j 0 for j 1, , n
n
n
such that j1 r j 1 and such that x
r a . We see that
j1 j j
n xy n n rjaj
j1 rjy
j1 rj aj y co A B . j1 c. Prove that if A and B are nonempty subsets of R k and x co A and y co B then
x y co A co B .
Choose a positive integer n and nonnegative numbers r j and members a j of the set A for
n
j 1, , n such that x
r a . Then
j1 j j
n xy n rjaj
j1 n rjy
j1 rj aj y .
j1 From part b we know that each number a j y must belong to co A B and since co A B is
convex we have
n xy rj aj y co A B . j1 Exercises on Metric Spaces
1. Suppose that X, d is a metric space and that we have defined
x, y d x, y if d x, y 1 1 if d x, y 1 . Prove that is also a metric on the set X.
The equation x, y y, x either says that d x, y d y, x or it says that 1 1. In either case, the
equation x, y y, x is true.
The equation x, y 0 holds if and only if d x, y 0 which holds if and only if x y.
Finally suppose that x, y and z belong to X. If either of the numbers x, y or y, z is equal to 1
then the inequality
x, z
x, y y, z
is assured. Otherwise 112 x, z
d x, z
d x, y d y, z x, y y, z
and once again the inequality x, z
x, y y, z holds.
2. Prove that if we define
d a, b x 1 x 2  y 1 y 2 
whenever a x 1 , y 1 and b x 2 , y 2 are points in R 2 then d is a metric on R 2 .
The fact that d a, b 0 if and only if a b is obvious and so is the equation d a, b d a, b for all
a and b. Now suppose that a x 1 , y 1 , b x 2 , y 2 and c x 3 , y 3 . We observe that
d a, c x 1 x 3  y 1 y 3 
x2 x2 x 1 x 2  x 2 x 1 x 3  y 1 y2 y2 x 3  y 1 y 2  y 2 y3 
y3  d a, b d b, c .
3. Prove that if we define
d a, b 2x 1 x 2  3y 1 y 2 
whenever a x 1 , y 1 and b x 2 , y 2 are points in R 2 then d is a metric on R 2 .
The solution of this exercise is virtually identical to that of Exercise 2.
4. Sketch the ball B O, 1 in R 2 with the Euclidean metric, with the Ýmetric and with each of the metric...
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 Fall '08
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 Math, Calculus

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