Leonard I. E. - Solutions Manual to Accompany Geometry of Convex Sets

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Library of Congress Cataloging-in-Publication Data:

Leonard, I. Ed., 1938

Geometry of convex sets / I.E. Leonard, J.E. Lewis.

pages cm

Includes bibliographical references and index.

ISBN 978-1-119-02266-4 (cloth) | ISBN 978-1-119-18418-8 (solutions manual)

1. Convex sets. 2. Geometry. I. Lewis, J. E. (James Edward) II. Title.

QA640.L46 2016

516.08dc23

2015021566

These are the solutions to the odd numbered problems in the text The Geometry of Convex Sets by I. E. Leonard and J. E. Lewis.

Some of the solutions are from assignments we gave in class, some are not. In all of the solutions, we have provided details that added to the clarity and ease of understanding for beginning students, and when possible to the elegance of the solutions.

Ed and Ted

Edmonton, Alberta, Canada

March 2016

A remark about the exercises is necessary. Certain questions are phrased as statements to avoid the incessant use of prove that. See Problem 1, for example. Such statements are supposed to be proved. Other questions have a truefalse or yesno quality. The point of such questions is not to guess, but to justify your answer. Questions marked with are considered to be more challenging. Hints are given for some problems. Of course, a hint may contain statements that must be proved.

1. Let be a nonempty set in . If every three points of are collinear, then is collinear.

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   Solution

   Let and be two distinct points in , then there is a unique line passing through these two points. Now let be an arbitrary point in , from the hypothesis, , and must be on some line , and since and uniquely determine the line , we must have . Therefore, every point in is on the line .

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2. Given that the line has the linear equation
   

   show that the point

   

   is on the line, and that the vector is parallel to the line.

   Hint. If is on the line and if is also on the line, then must be parallel to the line.

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   Solution

   Substituting the coordinates of this point into the linear equation for , we see that

   

   so that the given point is on