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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

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

 $KerT\subset Ker\left(T\cap H\right).$

Therefore

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

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

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

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

where

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

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

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