Ugrás a tartalomhoz

Convex Geometry

Csaba Vincze (2013)

University of Debrecen

8.4 Excercises

8.4 Excercises

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   a n d   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 = { ( 1 , - 2 ) }   a n d   E = { ( 4,1 ) , ( - 1,1 ) , ( 0 , - 1 ) }

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 = { ( 1 , - 2 ) , ( - 3,1 ) }   a n d   E = { ( 4,1 ) , ( - 1,1 ) }

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 = { ( 1 , - 2 ) , ( - 3,1 ) }   a n d   E = { ( 4,1 ) , ( - 1,1 ) , ( 0 , - 1 ) }

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 = { ( - 1,1 , 1 ) , ( 1,1 , - 1 ) , ( 1 , - 1,1 ) , ( 0,0 , 3 ) }

and

E = { ( 1,2 , 5 ) , ( 1 , - 2,3 ) }

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

( 1,1 ) , ( 2,3 ) , ( 3,2 ) , ( 4,3 ) .

Find the best affine approximation to all the points.

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

( 1,1 , 1 ) , ( 2,3 , - 1 ) , ( 3 , - 2,1 ) , ( - 1,1 , 2 ) .

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