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? *