1873_solutions

Set h is convex interpret this fact geometrically

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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  2|x 1 x 2 |  3|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 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 Ý-metric and with each of the metric...
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