Csaba Vincze (2013)

University of Debrecen

**Excercise 3.4.1 **
*Let D be the set consisting of the elements *

${v}_{1}=\left(\mathrm{1,0}\right),{v}_{2}=\left(\mathrm{1,3}\right),{v}_{3}=\left(\mathrm{4,3}\right),{v}_{4}=\left(\mathrm{4,0}\right)$ |

in the coordinate plane. Find a Radon's partition for D.

**Excercise 3.4.2 **
*Let D be the set consisting of the elements *

${v}_{1}=\left(\mathrm{1,1}\right),{v}_{2}=\left(\mathrm{4,1}\right),{v}_{3}=\left(\mathrm{5,2}\right),{v}_{4}=\left(\mathrm{2,3}\right),{v}_{5}=\left(\mathrm{2,2}\right)$ |

in the coordinate plane. Find a Radon's partition for D.

**Excercise 3.4.3 **
*Let D be the set consisting of the elements *

${v}_{1}=(\mathrm{2,0},-1),{v}_{2}=(\mathrm{1,1},2),{v}_{3}=(0,-\mathrm{1,1}),{v}_{4}=(-\mathrm{1,0},0),$ |

${v}_{5}=(\mathrm{1,0},1),{v}_{6}=(0,-\mathrm{3,3})$ |

in the coordinate space of dimension 3. Find a Radon's partition for D.

**Excercise 3.4.4 **
*Prove the one-dimensional version of Helly's theorem. *

Hint. Use that real numbers form an Archimedean complete totally ordered field.

**Excercise 3.4.5 **
*Let B be the collection consisting of convex subsets *

${B}_{1},\mathrm{\dots},{B}_{k}$ |

in the coordinate space of dimension n. Prove that if k is at least n and every subfamily of n sets in B has a non-empty intersection then the family of all sets in B has a common transversal parallel to any given 1-dimensional affine subspace/line in the space.

Hint. Let a 1-dimensional affine subspace be given and consider its orthogonal complement of dimension n - 1. Use Helly's original theorem3.3.1 to find a common point for the projected sets.

**Excercise 3.4.6 **
*How to generalize Corollary 3.3.4 to the coordinate space of dimension n? *

**Excercise 3.4.7 **
*Why Corollary 3.3.4 is a special case of Klee's second theorem? *

**Excercise 3.4.8 **
*How to generalize Jung's theorem 3.3.5 to the coordinate space of dimension three? *

**Excercise 3.4.9 **
*How to generalize inequality 3.16 to the coordinate space of dimension three? *