Ugrás a tartalomhoz

Convex Geometry

Csaba Vincze (2013)

University of Debrecen

6.2 Excercises

6.2 Excercises

Excercise 6.2.1 Prove that the star-shaped and convex sets of the coordinate line of dimension 1 coincide.

Excercise 6.2.2 Prove the one-dimensional version of Klee's theorem 4.1.6.

Excercise 6.2.3 Prove that we can apply the inductive hypothesis to the family 6.7.

Hint. Taking a countable subfamily

B 1 * K H , , B m * K H ,

(6.13)

we have that the intersection of the members in 6.13 is

m = 1 B m * K H = T H ,

where T is the intersection of the countable family

B 1 , , B m , , B 1 * , , B m * , .

Because of our assumptions T is a star-shaped set and, by the second step, its kernel is contained in H. This means that the intersection of T and H is also a star-shaped set and

K e r T K e r ( T H ) .

Therefore

d i m K e r ( T H ) d i m K e r T k

and we can use the inductive hypothesis as mentioned above.

Excercise 6.2.4 Prove that we can apply the inductive hypothesis to the family 6.12.

Let k be an arbitrary natural number and

T k : = { ( x , y ) | x 0 , k y 0 } { ( x , y ) | x k , y k } .

(6.14)

Figure 40: Excercise 6.2.5.

Figure 41: Excercise 6.2.5.

Figure 42: The intersection of T(1) and T(2) (left). The kernel (right).

Figure 43: Excercise 6.2.6.

Excercise 6.2.5 Prove that every finite subfamily of 6.14 has a star-shaped intersection whose kernel is of dimension two but the intersection of all thethe sets is not star-shaped.

Let k be an arbitrary natural number and

D k : = ( c o n v { a , b , c k } \ s ( a , b ) ) { a , b } ,

(6.15)

where

a = ( - 1,0 ) , b = ( 1,0 )   a n d   c k = ( 0,1 / k ) .

Excercise 6.2.6 Prove that every finite subfamily of 6.15 has a star-shaped intersection whose kernel is of dimension two but the intersection of all the sets is not star-shaped.

Hint. Especially

k = 1 D k = { a , b } .

Remark The excercise illustrates that the condition for the countable subfamilies in theorem 1.1 could not be weakened to a finite version.