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## Convex Geometry

Csaba Vincze (2013)

University of Debrecen

3.4 Excercises

## 3.4 Excercises

Excercise 3.4.1 Let D be the set consisting of the elements

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

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

Excercise 3.4.2 Let D be the set consisting of the elements

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

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

Excercise 3.4.3 Let D be the set consisting of the elements

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

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

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?