1873_solutions

A b c a b c b c b c 0 the point x is shown to

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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...
View Full Document

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.

Ask a homework question - tutors are online