Csaba Vincze (2013)

University of Debrecen

**Excercise 8.4.1 **
*Let D be the set of vertices of a square in the coordinate plane. Find the sets *

${D}_{1},{D}_{2}\text{}and\text{}{D}_{3}.$ |

**Excercise 8.4.2 **
*Find an example to show that the Kirchberger number n+2 can not be reduced. *

Hint. The pairs of points on the same diagonals of a square can not be separated by a line but any three of them can be strictly separated.

**Excercise 8.4.3 **
*Prove or disprove that the sets *

$D=\left\{\right(1,-2\left)\right\}\text{}and\text{}E=\left\{\right(\mathrm{4,1}),(-\mathrm{1,1}),(0,-1\left)\right\}$ |

can be separated by a line in the coordinate plane. Find the equation of the separating line if exists.

**Excercise 8.4.4 **
*Prove or disprove that the sets *

$D=\left\{\right(1,-2),(-\mathrm{3,1}\left)\right\}\text{}and\text{}E=\left\{\right(\mathrm{4,1}),(-\mathrm{1,1}\left)\right\}$ |

can be separated by a line in the coordinate plane. Find the equation of the separating line if exists.

**Excercise 8.4.5 **
*Prove or disprove that the sets *

$D=\left\{\right(1,-2),(-\mathrm{3,1}\left)\right\}\text{}and\text{}E=\left\{\right(\mathrm{4,1}),(-\mathrm{1,1}),(0,-1\left)\right\}$ |

can be separated by a line in the coordinate plane. Find the equation of the separating line if exists.

**Excercise 8.4.6 **
*Prove or disprove that the sets *

$D=\left\{\right(-\mathrm{1,1},1),(\mathrm{1,1},-1),(1,-\mathrm{1,1}),(\mathrm{0,0},3\left)\right\}$ |

and

$E=\left\{\right(\mathrm{1,2},5),(1,-\mathrm{2,3}\left)\right\}$ |

can be separated by a plane in the coordinate space of dimension three. Find the equation of the separating plane if exists.

**Excercise 8.4.7 **
*Find the error for the best affine approximation to each subset of three points in *

$\left(\mathrm{1,1}\right),\left(\mathrm{2,3}\right),\left(\mathrm{3,2}\right),\left(\mathrm{4,3}\right).$ |

Find the best affine approximation to all the points.

**Excercise 8.4.8 **
*Find the best affine approximation of the set of points *

$(\mathrm{1,1},1),(\mathrm{2,3},-1),(3,-\mathrm{2,1}),(-\mathrm{1,1},2).$ |

**Excercise 8.4.9 **
*How to generalize the best approximation problem and the solution to the coordinate space of dimension n? *