Ugrás a tartalomhoz

## Convex Geometry

Csaba Vincze (2013)

University of Debrecen

2.4 Excercises

## 2.4 Excercises

Excercise 2.4.1 Check the affine dependence/independence of the system

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

Excercise 2.4.2 Prove or disprove the following statements:

• Linearly independent systems are affinely independent.

• Affinely independent systems are linearly independent.

• Linearly dependent systems are affinely dependent.

• Affinely dependent systems are linearly dependent.

Excercise 2.4.3 Check the affine dependence/independence of the system

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

Excercise 2.4.4 Check the affine dependence/independence of the following systems of vectors.

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

 ${v}_{4}=\left(1,2,3,4\right),{v}_{5}=\left(0,1,2,3\right).$

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

 ${v}_{4}=\left(2,9,3,5\right).$

 ${v}_{1}=\left(-3,1,5,3,2\right),{v}_{2}=\left(2,3,0,1,0\right),{v}_{3}=\left(1,2,3,2,1\right),$

 ${v}_{4}=\left(3,-5,-1,-3,-1\right),{v}_{5}=\left(3,0,1,0,0\right).$

Excercise 2.4.5 Find the affine coordinates of

in the coordinate plane with respect to

 ${v}_{1}=\left(2,0\right),{v}_{2}=\left(0,5\right),{v}_{3}=\left(-1,1\right).$ (2.14)

Using affine coordinates how to characterize points in the interior, points on the boundary or points outside of conv 2.14.

Excercise 2.4.6 Prove that

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

 ${v}_{4}=\left(1,0,3,1\right),{v}_{5}=\left(-1,1,0,1\right)$

are affinely independent and find the affine coordinates of the origin in the coordinate space of dimension four.

Excercise 2.4.7 Consider the vector

 $v=\frac{1}{2}{v}_{1}+\frac{1}{4}{v}_{2}+\frac{1}{6}{v}_{3}+\frac{1}{12}{v}_{4},$

where

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

Use the procedure in the proof of Carathéodory's theorem 2.2.1 to reduce the number of the members in the convex combination as far as possible.

Excercise 2.4.8 Consider the vector

 $v=\frac{1}{5}{v}_{1}+\frac{1}{5}{v}_{2}+\frac{1}{5}{v}_{3}+\frac{1}{5}{v}_{4}+\frac{1}{5}{v}_{5},$

where

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

Use the procedure in the proof of Carathéodory's theorem 2.2.1 to reduce the number of the members in the convex combination as far as possible.

Excercise 2.4.9 Consider the vector

 $v=\frac{1}{10}{v}_{1}+\frac{3}{20}{v}_{2}+\frac{1}{4}{v}_{3}+\frac{1}{5}{v}_{4}+\frac{2}{25}{v}_{5}+\frac{11}{50}{v}_{6},$

where

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

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

Use the procedure in the proof of Carathéodory's theorem 2.2.1 to reduce the number of the members in the convex combination as far as possible.

Excercise 2.4.10 Consider the vector

 $v=\frac{1}{24}{v}_{1}+\frac{1}{12}{v}_{2}+\frac{1}{8}{v}_{3}+\frac{5}{12}{v}_{4}+\frac{1}{3}{v}_{5},$

where

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

 ${v}_{5}=\left(1,0,1\right).$

Use the procedure in the proof of Carathéodory's theorem 2.2.1 to reduce the number of the members in the convex combination as far as possible.