Real Algebraic Geometry II – Exercise Sheet 7

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

Summer Term 2017 Markus Schweighofer

Real Algebraic Geometry II – Exercise Sheet 7

Exercise 1(4P) LetKbe a subfield ofR,V a finite-dimensionalK-vector space,A

⊆

V convex andF a maximal nontrivial face ofA. Show that Fis exposed.

Exercise 2(8P) Let n

∈

_N0 andd

∈

_Nbe even,V

⊆

_R

[

X₁, . . . ,X_n

]

theR-vector space of alld-forms innvariables. LetP

⊆

Vbe the cone of all positive semidefinited-forms ofnvariables. Show:

(a) Pis closed.

(b) P^◦ consists exactly of the positive definited-forms innvariables.

(c) For everyx

∈

_Rⁿ

\ {

0

}

the set F_x:

= {

f

∈

P

|

f

(

x

) =

0

}

is a maximal non-trivial face ofP.

(d) For every maximal non-trivial faceF ofV there exists anx

∈

_Rⁿ

\ {

0

}

such that F

=

Fx.

Exercise 3 (8P) Suppose K is a subfield of R, n

∈

_{N0 and} V is an n-dimensional topological K-vector space. Let A

⊆

V be a convex set and x

∈

V

\

A. Show that there existK-linear functionsϕ₁, . . . ,ϕ_n:V

→

_Rsuch that for everyy

∈

A, there exists j

∈ {

1, . . . ,n

}

satisfying

ϕ₁

(

x

) =

ϕ₁

(

y

)

, . . . ,ϕ_j−1

(

x

) =

ϕ_j−1

(

y

)

andϕ_j

(

x

) <

ϕ_j

(

y

)

.

Exercise 4(4P)

(a) Prove or disprove the following: For anyn

∈

_N0 and closedA

⊆

_Rⁿ, conv

(

A

)

is also closed.

(b) Findn

∈

_Nand two nonempty disjoint convex sets A,B

⊆

_Rⁿ such that there exists no linear function ϕ : Rⁿ

→

_R satisfying ϕ

(

a

) <

ϕ

(

b

)

for all a

∈

A and b

∈

B.

Please submit until Tuesday, June 13, 2017, 9:55 in the box named RAG II near to the room F411.