Csaba Vincze (2013)

University of Debrecen

**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

${B}_{1}^{\mathrm{*}}\cap K\cap H,\mathrm{\dots},{B}_{m}^{\mathrm{*}}\cap K\cap H,\mathrm{\dots}$ |
(6.13) |

we have that the intersection of the members in 6.13 is

$\bigcap _{m=1}^{\mathrm{\infty}}{B}_{m}^{\mathrm{*}}\cap K\cap H=T\cap H,$ |

where T is the intersection of the countable family

${B}_{1},\mathrm{\dots},{B}_{m},\mathrm{\dots},{B}_{1}^{\mathrm{*}},\mathrm{\dots},{B}_{m}^{\mathrm{*}},\mathrm{\dots}.$ |

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

$KerT\subset Ker(T\cap H).$ |

Therefore

$\mathrm{d}\mathrm{i}\mathrm{m}Ker(T\cap H)\ge \mathrm{d}\mathrm{i}\mathrm{m}KerT\ge k$ |

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

${T}_{k}:=\left\{\right(x,y\left)\right|x\ge 0,k\ge y\ge 0\}\cup \{(x,y)|x\ge k,y\ge k\}.$ |
(6.14) |

**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

${D}_{k}:=\left(conv\right\{a,b,{c}_{k}\}\backslash s(a,b\left)\right)\cup \{a,b\},$ |
(6.15) |

where

$a=(-\mathrm{1,0}),b=\left(\mathrm{1,0}\right)\text{}and\text{}{c}_{k}=(\mathrm{0,1}/k).$ |

**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

$\bigcap _{k=1}^{\mathrm{\infty}}{D}_{k}=\{a,b\}.$ |

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