Csaba Vincze (2013)
University of Debrecen
Excercise 4.4.1 Prove the one-dimensional version of Helly's theorem for an arbitrary collection of compact intervals.
Hint. Use that real numbers form an Archimedean complete totally ordered field.
Excercise 4.4.2 Let B be the family of compact convex sets in the coordinate space of dimension n and suppose that B contains at least n members. Prove that if every subfamily of n sets in B has a non-empty intersection then the familyof all sets in B has a common transversal parallel to any given 1-dimensional affine subspace/line in the space.
Hint. Let an 1-dimensional affine subspace be given and consider its orthogonal complement of dimension n - 1. Use the general version 4.1.2 of Helly's theorem to find a common point for the projected sets.
Excercise 4.4.3 Prove theorem 4.1.5.
Excercise 4.4.4 Prove the general version of Jung's theorem 4.3.5 to find the smallest radius for a universal covering disk in the plane.
Excercise 4.4.5 Calculate the perimeter and the area of a Reuleaux triangle in terms of the side of the equilateral triangle.
Excercise 4.4.6 Find the measure of the interior angle at the corners of the Reuleaux triangle.
Excercise 4.4.7 Prove that Reuleaux triangles are complete in the sense that no points from their complements can be added to them without increasing the diameter.
Excercise 4.4.8 How to generalize theorem 4.2.2 to the coordinate space of dimension three?