Exercise 6: [Challenge] Suppose that you have a covering of a sphere by finitely many (closed) hemispheres. Prove that there is a subcollection of less than five hemispheres that cover the sphere. Remark: A set is called closed if it contains its boundary points. In the case of a hemisphere, this means it contains the boundary (great) circle. Compare to a closed interval, e.g., the interval 0, 1] is closed since it contains the boundary points of 0 and 1. Last modified: October 2, 2019, Due: October 9, 2019. 1 Hint 1: Consider the case of five hemispheres. Prove that four hemispheres suffice. Hint 2: Consider the case of a circle: Show that if four semicircles cover a circle, then three of them suffice.

Exercise 6: [Challenge] Suppose that you have a covering of a sphere by finitely many (closed) hemispheres. Prove that there is a subcollection of less than five hemispheres that cover the sphere. Remark: A set is called closed if it contains its boundary points. In the case of a hemisphere, this means it contains the boundary (great) circle. Compare to a closed interval, e.g., the interval 0, 1] is closed since it contains the boundary points of 0 and 1. Last modified: October 2, 2019, Due: October 9, 2019. 1 Hint 1: Consider the case of five hemispheres. Prove that four hemispheres suffice. Hint 2: Consider the case of a circle: Show that if four semicircles cover a circle, then three of them suffice.

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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