Real Algebraic Geometry II – Exercise Sheet 8

(1)

Universität Konstanz Tom-Lukas Kriel Department of Mathematics and Statistics María López Quijorna

Summer Term 2017 Markus Schweighofer

Real Algebraic Geometry II – Exercise Sheet 8

Exercise 1 (8P) Let f: R

→

_R be a semialgebraic function and R be a real closed extension field ofR.

(a) Show that Transfer_R,R

(

_Γ_f

) ⊆

R²equals the graph Γg of anR-semialgebraic func- tiong: R

→

R.

(b) Let x

∈

_R and δ

∈

m_R

\ {

0

}

. Show that f is continuous at x if and only if g

(

x

−

δ

)

,g

(

x

+

δ

) ∈

_O_R and

st

(

g

(

x

−

δ

)) =

f

(

x

) =

st

(

g

(

x

+

δ

))

.

(c) Let x

∈

_R, δ

∈

m_R

\ {

0

}

and a

∈

R. Show that f is differentiable at x with f⁰

(

x

) =

aif and only if ^g⁽^x⁻^δ₋⁾⁻_δ ^g⁽^x⁾,^g⁽^x⁺^δ_δ⁾⁻^g⁽^x⁾

∈

_O_R and

st

g

(

x

−

δ

) −

g

(

x

)

−

δ

=

a

=

st

g

(

x

+

δ

) −

g

(

x

)

δ

.

Exercise 2(4P) Let A

⊆

_Rⁿ be closed and convex. Show that the following are equiv- alent forx

∈

A:

(a) x is an extreme point of A.

(b) For everyε

>

0, there exists a linear function ϕ: Rⁿ

→

_Rsuch that ϕ

(

x

) >

cand

∀

y

∈

A:

(

ϕ

(

y

) >

c

= ⇒ k

x

−

y

k <

ε

)

.

Exercise 3(4P) Let A

⊆

_Rⁿbe convex. An exposed extreme pointof Ais a pointx

∈

A such that

{

x

}

is an exposed face ofA. Now suppose that Ais compact. Show that the closure of the set of exposed extreme points of Acontains all extreme points of A.

Hint:Considerx

∈

extrA. Letε

>

0 be arbitrary. We want to find an exposed extreme point z of Awith

k

z

−

x

k <

ε. For this purpose, choosew

∈

_Rⁿ andc

∈

_R such that w^Tx

>

cand

∀

y

∈

A :

(

w^Ty

>

c

= ⇒ k

x

−

y

k <

ε

)

. Now show that for λ

∈

_R big enough, every pointz

∈

Amaximizing

k

z

− (

x

−

λw

)k

does the job.

Please submit until Tuesday, June 20, 2017, 9:55 in the box named RAG II near to the room F411.