Table Of ContentUPPER SEMILATTICES OF FINITE-DIMENSIONAL GAUGES
7
S.S.KUTATELADZE
0
0
In Memory of Alex Rubinov (1940–2006)
2
n
a Abstract. This is a brief overview of some applications of the ideas of ab-
J stractconvexity totheuppersemilatticesofgauges infinitedimensions.
2
]
A
F Introduction
.
h
t Duality in convexity is a simile of reversal in positivity. The ghosts of this
a
similarity underlay the research on abstract convexity we were engrossed in with
m
AlexRubinovintheearly1970s. Oureffortsledtothesurvey[1]anditsexpansion
[
inthe namesakebook[2]. We alwayscherisheda hope to revisitthis areaandshed
1 light on a few obscurities. However, the fate was against us.
v Inspecting the archive of our drafts of these years, I encountered several items
9 on the cones of Minkowski functionals or, equivalently, gauges. The results on
5
the Minkowski duality in finite dimensions are practically unavailable in full form,
0
whereasthey restonthe technique that is still uncommon andunpopular but defi-
1
0 nitely profitable. The theorems ongaugesappearedmostly insome mimeographed
7 local sources that had disappeared two decades ago. We hoped and planned to
0 expatiate on these matters when time will come.
/
h Alex Rubinov was my friend up to his terminal day. He shared his inspiration
t and impetus with me. So does and will do his memory...
a
m An abstract convex function is the upper envelope of a family of simple func-
tions [1]–[3]. The cone of abstract convex elements is an upper semilattice. We
:
v describe the bipolar of such a semilattice through majorization generated by its
i
X polar. Polyhedral approximation simplifies the generatorsof the polar in finite di-
mensions to discrete measures. Decomposition reduces the matter to Jensen-type
r
a inequalities, which opens a possibility of linear programming and we are done.
These ideas characterize our approach.
This article is organized as follows: Section 1 is a short discussion of majoriza-
tion and decomposition in the spaces of continuous functions. Section 2 addresses
the space of convex sets in finite dimensions and the influence of polyhedral ap-
proximation on the structure of dual cones. Section 3 illustrates the use of linear
programming for revealing continuous linear selections over convex figures. Sec-
tion 4 collects some dual representations for the members of upper semilattices of
gauges. In Section 5 we deal with some upper lattices of gauges that are closed
under intersection.
Date:December31,2006.
Key words and phrases. Majorization,Minkowskifunctional,abstractconvexity.
1
2 S.S.KUTATELADZE
1. Majorization and Decomposition
It was long ago in 1954 that Reshetnyak suggested in his unpublished thesis [4]
to compare (positive) measures on the Euclidean unit sphere S as follows:
N−1
1.1. A measure µ (linearly) majorizes or dominates a measure ν provided that
to each decomposition of S into finitely many disjoint Borel sets U ,...,U
N−1 1 m
there are measures µ ,...,µ with sum µ such that every difference µ −ν|
1 m k Uk
annihilates all restrictions to S of linear functionals over N. In symbols, we
N−1
R
write µ≫ ν.
N
ReshetnyRak proved that
pdµ≥ pdν
Z Z
SN−1 SN−1
for every sublinear functional p on N if µ ≫ ν. This gave an important trick
N
forgeneratingpositivelinearfunctioRnalsovervaRriousclassesofconvexsurfacesand
functions.
1.2. A similar idea was suggested by Loomis [5] in 1962 within Choquet theory.
Ameasureµaffinelymajorizesameasureν,bothgivenonacompactconvexsubset
QofalocallyconvexspaceX,providedthattoeachdecompositionofν intofinitely
many summands ν ,...,ν there are measures µ ,...,µ with µ such that every
1 m 1 m
differenceµ −ν annihilatesallrestrictionstoQoftheaffine functionsoverX. In
k k
symbols,µ≫ ν. Many applications of affine majorizationare set forthin [6].
Aff(Q)
Cartier, Fell, and Meyer proved in [7] that
fdµ≥ fdν
Z Z
Q Q
for every continuous convex function f on Q if and only if µ≫ ν.
Aff(Q)
An analogous necessity part for linear majorization was published in [8]. In
applications we use a more detailed version of majorization [9]:
1.3. Decomposition Theorem. Assume that H ,...,H are cones in a vector
1 n
lattice X. Assume further that f and g are positive linear functionals on X. The
inequality
f(h ∨···∨h )≥g(h ∨···∨h )
1 n 1 n
holds for all h ∈H (k :=1,...,n) if and only if to each decomposition of g into
k k
asumofnpositivetermsg =g +···+g thereisadecompositionoff intoasum
1 N
of n positive terms f =f +···+f such that
1 n
f (h )≥g (h ) (h ∈H ; k :=1,...,n).
k k k k k k
2. The Space of Convex Figures
We will proceed in the Euclidean space N.
R
2.1. A convex figure is a compact convex set. A convex body is a solid convex
figure. The Minkowski duality identifies a convex figure S in N with its support
R
function S(z):=sup{(x,z)|x∈S} for z ∈ N. Considering the members of N
as singletons, we assume that N lies in theRset V of all compact convex subsRets
N
R
of N.
R
UPPER SEMILATTICES OF FINITE-DIMENSIONAL GAUGES 3
2.2. The Minkowski duality makes V into a cone in the space C(S ) of con-
N N−1
tinuousfunctionsontheEuclideanunitsphereS ,theboundaryoftheunitball
N−1
z . Thisyieldsistheso-calledMinkowski structure onV . Additionofthesupport
N N
functions of convex figures amounts to taking their algebraic sum, also called the
Minkowski addition. It is worth observing that the linear span [V ] of V is dense
N N
in C(S ), bears a natural structure of a vector lattice and is usually referred
N−1
to as the space of convex sets. The study of this space stems from the pioneering
breakthroughof Alexandrov [10] in 1937 and the further insights of Radstro¨m [11]
and Ho¨rmander [12].
2.3. A gauge p is a positive sublinear functional on a real vector space X viewed
as the Minkowski functional of the conic segment S := {p ≤ 1} := {x ∈ X |
p
p(x)≤1}. The latter is also referred to as a gauge or caliber. A gauge p is a norm
providedthatitsball S issymmetricandabsorbing. Recallthatthesubdifferential
p
orsupportset ∂pofpisthedualball orpolar ofS . ThepolarofaballS isdenoted
p
by S◦ and the dual norm of k·kS is k·kS◦. The “donkey bridge” of functional
analysis consists in the duality rules:
k·kS =S◦(·), k·kS◦ =S(·).
We will restrict exposition to the norms and balls of N by way of tradition.
R
2.4. ApproximationLemma. IfH isasubconeofV thenthesignedmeasures
N
with finite support are sequentially weakly* closed in the dual cone H∗.
Proof. Let µ∈H∗. The mappings
z 7→µ (z); z 7→µ (z),
+ −
with z ∈ N, are linear functionals on N. Therefore, there are u,v ∈ N such
R R R
that µ (z)=(u,z) and µ (z)=(v,z). Put
+ −
µ ;=µ +mes+|u|ε ;
1 + −u/|u|
µ :=µ +mes+|v|ε ;
2 − −v/|v|
µ :=µ +|v|ε ; µ :=µ +|u|ε .
1 1 −v/|v| 2 2 −u/|u|
As usual, ε is the Dirac measure at z ∈ N, while |·| is the Euclidean norm on
z
R
N, and mes is the Lebesgue measure on S : i. e. the surface area function of
N−1
R
the Euclidean ball z := {x ∈ N : |x| ≤ 1}. Note that µ = µ −µ . Moreover,
N 1 2
R
themeasuresµ andµ arenondegenerateandtranslation-invariant. Indeed,check
1 2
thatsoisµ . Thissignedmeasureisclearlypositiveandnotsupportedbyanygreat
1
hypersphere. We are left with validating translation-invariance. If k := 1,...,N
then
e dµ = e dµ + e dµ(z )−(u,e )=(u,e )−(u,e )=0.
Z j 1 Z j + Z j N k k k
SN−1 SN−1 SN−1
Consideraconvexfigurexwhosesurfaceareafunctionµ(x)equalsµ . Theexistence
1
of this figure is guaranteed by the celebrated Alexandrov Theorem [10, p.108].
Let (x ) be a sequence of polyhedra including x and converging to x in the
m
Hausdorff metric on [V ]which is induced by the Chebyshev norm on C(S ).
N N−1
4 S.S.KUTATELADZE
Then the measures µ1 = µ(x ) converge weakly* to µ and µ1 ≫ µ . Indeed,
m m 1 m n 1
given a convex figure z, we have R
zdµ1 = zdµ(x )=nV(z,x ,...,x )
Z m Z m m m
SN−1 SN−1
≥nV(z,x,...,x)= zdµ(x)= zdµ
Z Z 1
SN−1 SN−1
by the inclusion monotonicity of the mixed volume V(·,...,·) in every argument..
By analogy, there is a sequence (µ2 ), converging weakly* to µ and such that
m 2
µ2 ≫ µ . Putting
m n 2
R µ1 :=µ1 +|v|ε ;
m m −v/|v|
µ2 :=µ2 +|u|ε ,
m m −u/|u|
we see that µ1 −µ2 converges weakly* to µ. The proof is complete.
m m
3. Labels and Decompositions
TheApproximationLemmaallowsustoreduceconsiderationtosignedmeasures
with finite support. These measures decompose easily. We will exhibit a typical
application.
3.1. A family (µ ,...,µ ) of regular Borel measures on the sphere S is a la-
1 n N−1
beling on N provided that (µ (x), ..., µ (x)) ∈ x for all x ∈ V . The vector
1 n N
R
(µ (x),...,µ (x)) is a label of x.
1 n
3.2. Proposition. A family (µ ,...,µ ) is a labeling on N if and only if
1 n
R
n
ε − x µ ∈V∗.
x k k N
Xk=1
for all x∈S .
N−1
Proof. The Minkowski duality is an isomorphism of the relevant structures.
Hence, the definition of labeling can be rephrased as follows:
n
x µ (x)≤x(x) (x∈ n, x∈V ).
k k N
R
Xk=1
3.3. UsinglinearmajorizationfordescribingV∗, wecanwrite downsomecriteria
N
for labeling in terms of decompositions. For simplicity, we will argue in the planar
case.
Consider the conditions:
(++) ε(△1,△2)+△1 µ−1+△2 µ−2 ≫2 △1 µ+1+△2 µ+2;
R
(+−) ε(△1,−△2)+△1 µ−1+△2 µ+2 ≫2 △1 µ+1+△2 µ−2;
R
(−+) ε(−△1,△2)+△1 µ+1+△2 µ−2 ≫2 △1 µ−1+△2 µ+2;
R
(−−) ε(−△1,−△2)+△1 µ+1+△2 µ+2 ≫2 △1 µ−1+△2 µ−2;
R
UPPER SEMILATTICES OF FINITE-DIMENSIONAL GAUGES 5
with (△ ,△ )∈S ∩ 2. Clearly, the requirement of 4.1 amounts to the four con-
1 2 1 R+
ditions simultaneously. By wayof example, we will elaborate the relevant criterion
only in the case of (+−).
3.4. Proposition. For (+−) to hold it is necessary and sufficient that to all
(△ ,△ )inS ∩ 2 andalldecompositions{(µ+) ,...,(µ+) }ofµ+ andaldecom-
1 2 1 R+ 1 1 1 m 1
positions{(µ−) ,...,(µ−) }ofµ−thereexistadecomposition{(µ−) ,...,(µ−) }
2 1 2 m 2 1 1 1 m
of µ−, a decomposition {(µ+) ,...,(µ+) } of µ+, and reals α ,...,α that make
1 2 1 2 m 2 1 m
compatible the simultaneous inequalities:
α ≥0;...;α ≥0;α +...+α =1;
1 m 1 m
△ (x −x +α e )
1 (µ−1)k (µ+1)k k 1
=△ (x −x +α e ) (k :=1,...,m),
2 (µ−2)k (µ+2)k k 2
where x is the representing point of µ; i. e., µ(u)=(u,x ) for all u∈ 2.
µ µ
Proof. ⇐=: Let(△ ,△ )∈S ∩ 2 andlet{ν ,...,ν }beanarbitrRarydecom-
1 2 1 R+ 1 m
position of △ µ++ △ µ−. By the Riesz Decomposition Lemma there are a de-
1 1 2 2
composition {(µ+) ,...,(µ+) } of µ+ and a decomposition {(µ−) ,...,(µ−) }
1 1 1 m 1 2 1 2 m
of µ− such that △ (µ+) + △ (µ−) = ν . Find some parameters satisfying the
2 1 1 k 2 2 k k
simultaneous inequalities and put
µk :=△1 (µ−1)k+△2 (µ+2)k+αkε(△1,−△2).
Clearly, µ ≥0 and, moreover,
k
m
µk =△1 µ−1+△2 µ+2 +ε(△1,−△2).
Xk=1
Furthermore,
x −x =△ x +△ x +α △ e
µk νk 1 (µ−1)k 2 (µ+2)k k 1 1
−α △ e −△ x −△ x =0,
k 2 2 1 (µ+1)k 2 (µ−2)k
and so µ −ν belongs to the polar of 2 in C(S ).
k k 1
R
=⇒: Assume (+−) valid.
Givendecompositions{(µ+) ,...,(µ+) }and{(µ−) ,...,(µ−) }thereisade-
1 1 1 m 2 1 2 m
composition {ν1,...,ν2m} of ε(△1,−△2)+△1 µ−1+△2 µ+2 such that
x =x ; x =x (k :=1,...,m).
νk (µ+1)k νm+k (µ−2)k
We are left with appealing to the Riesz Decomposition Lemma and represent-
ing the decomposition {ν ,...,ν } through the corresponding decompositions
1 2m
of ε(△1,−△2), △1 µ−1, and △2 µ+2. The proof is complete.
3.5. Ifitispossibletochosedecompositionsin3.4independentlyof(△ ,△ ),then
1 2
we come to a sufficient condition for labeling. Let us illustrate this by exhibiting
an example of one of the simplest labelings.
We will seek a labeling of the form
µ :=|µ+|ε −|µ−|ε ;
1 µ+/|µ+| µ−/|µ−|
µ :=|ν+|ε −|ν−|ε ,
2 ν+/|ν+| ν−/|ν−|
6 S.S.KUTATELADZE
with µ+, µ−, ν+, and ν− some points on the plane. The sufficient condition we
have just suggested paraphrases as follows:
α ,β ,a ,b ,γ ,c ≥0;
k k k k k k
α +a =1; β +b =1; γ +c =1 (k :=1,...,4);
k k k k k k
µ+ =α µ−+γ e ; β ν−+γ e =0;
1 1 1 1 1 2
ν+ =b ν−+c e ; a µ−+c e =0;
1 1 2 1 1 1
µ− =α µ+−γ e ; β ν−+γ e =0;
2 2 1 2 2 2
ν+ =b ν−+c e ; a µ+−c e =0;
2 2 2 2 2 1
µ− =α µ+−γ e ; β ν+−γ e =0;
3 3 1 3 3 2
ν− =b ν+−c e ; a µ+−c e =0;
3 3 2 3 3 1
µ+ =α µ +γ e ; β ν+−γ e =0;
4 − 4 1 4 4 2
ν− =b ν+−c e ; a µ−+γ e =0.
4 4 2 4 4 1
The solution of the last system is given by the parameters:
α =b =0; β =a =1;
k k k k
1
γ =c = (k :=1,...,4).
k k
2
Moreover,
1 1 1 1
µ+ = e ; ν+ = e ; µ− =− e ; ν− =− e .
1 2 1 2
2 2 2 2
Therefore,thesimplestlabelingofxisthepoint 1(x(e )−x(−e ),x(e )−x(−e )). It
2 1 1 2 2
is worth emphasizing that the validation of the above conditions belongs to linear
programming which enables us to seek for arbitrary labelings by signed measures
with finite support.
4. The Case of Joining Gauges
We now apply the above ideas to studying the classes of N-dimensional convex
surfaces which comprise upper semilattices in V . To simplify notation we will
N
discussonlyballs,denotingthesetofballsinV byVS . Itisconvenientformally
N N
toaddtheapextoVS . IfS ∈V S differsfromtheoriginthenweusethesymbol
N N
k·k not only for the gauge of S but also for the operator norm corresponding to
S
S in the endomorphism space L( N) of N. In other words,
R R
kxk :=inf{α>0|x/α∈S} (x∈ N);
S
R
kAk :=sup{kAxk |x∈S} (A∈L( N)).
S S
R
Recall that
S◦ ={x∈ N ||(x,y)|≤1 (y ∈S)},
R
where (·,·) is the standard inner product of N.
Observe that V S is a lattice and simultaRneously a cone. However, V S is not
N N
closed in V . This circumstance notwithstanding, given a family (S ) in V S,
N ξ ξ∈Ξ N
sometimeswemaysoundlyspeakoftheupperhull π↑(Ξ),lowerhull π (Ξ),andhull
↓
π(Ξ) of this family, implying the least closed cones that lie in V S, include S for
N ξ
allξ ∈Ξ,andareclosedunderthejoin,themeet,andbothoperationsinthelattice
UPPER SEMILATTICES OF FINITE-DIMENSIONAL GAUGES 7
of convex figures V . An example is provided by any instance of nondegenerate
N
family. The latter is by definition any family of nonzero sets (S ) such that,
ξ ξ∈Ξ
supkAk <+∞ (A∈L( N)).
Sξ R
ξ∈Ξ
Indeed, put
A(Ξ):={A∈L( N)|AS ⊂S (ξ ∈Ξ)},
ξ ξ
R
and let M(Ξ) be the set of the symmetric elements of V such that AS ⊂ S for
N
all A ∈ A(Ξ). Since (S ) is nondegenerate, all members of M(Ξ) but the zero
ξ ξ∈Ξ
singleton are absorbing. Moreover,M(Ξ) is clearly a closed sublattice of V .
N
We will need the helpful property of a nondegenerate family: If y ∈ N differs
R
from the zero of N then
R S
ξ
S :=
y
S (y)
ξ^∈Ξ ξ
is absorbing. Indeed, given z ∈ N we infer that
R
supSξ(y)Sξ◦(z)=supkykSξ◦kzkSξ =supky⊗zkSξ <+∞,
ξ∈Ξ ξ∈Ξ ξ∈Ξ
where y⊗z :x7→(y,x)z for all x∈ N. Hence, the polar of S is compact, which
y
R
implies that S is absorbing. Without further specification, we will address only
y
nondegenerate families of balls in the sequel.
4.1. Theorem. A gauge S belongs to π↑(Ξ) if and only if
S S
ξ
≤
n n
kxkkS◦ ξ_∈Ξ kxkkS◦
ξ
kP=1 kP=1
foranycollectionofthevectorsx ,...,x ∈ N thatarenotallzerosimultaneously.
1 p
R
Proof. It is obvious that π↑(Ξ) is the closure of the upper semilattice of all
H-convexfunctionswithH theconichullofthefamily(S ) . Thepolarofπ↑(Ξ)
ξ ξ∈Ξ
may be approximated with finitely supported signed measures by the Approxima-
tion Lemma. Using the Bipolar Theorem, we see that S ∈ π↑(Ξ) if and only if
n S(x ) ≥ S(y) whenever y,x ,...,x ∈ N satisfy n S (x ) ≥ S (y) for
k=1 k 1 n R k=1 ξ k ξ
aPll ξ ∈Ξ. By duality, S ∈π↑(Ξ) if and only if P
n n
kxkkSξ◦Sξ◦ ⊂ kxkkS◦S◦.
ξ^∈ΞXk=1 kX=1
Taking polars, we complete the proof of the theorem.
4.2. Corollary. A nonzero gauge S belongs to π↑(Ξ) if and only if
n
S
ξ
(4.2.1) S = S(x ) ,
k n
(x1,^...,xn)Xk=1 ξ_∈Ξ Sξ(xk)
kP=1
where the intersection ranges over all nonzero tuples (x ,...,x )∈( )N.
1 n
R
Proof. Clearly, (4.2.1) guarantees the inclusion of 4.1 and so S ∈ π↑(Ξ). The
last containment in turn implies the simple representation:
S
ξ
(4.2.2) S = S(x) .
S (x)
x^6=0 ξ_∈Ξ ξ
8 S.S.KUTATELADZE
Indeed, denote by S the right-hand side of (4.2.2). By 4.1, S ≤S. If z ∈ n then
R
S
e ξ e
S(z)= S(x) (z)
(cid:18) S (x)(cid:19)
x^6=0 ξ_∈Ξ ξ
e
S S (z)
ξ ξ
≤S(z) (z)=S(z) =S(z).
(cid:18) S (z)(cid:19) S (z)
ξ_∈Ξ ξ ξ_∈Ξ ξ
≈
Bythe MinkowskidualityS ≤S. DenotebyS theright-handsideof(4.2.1). Since
≈ ≈
S ≤S ≤S ≤S; therefore,eS =S and we are done.
4.3. Frome 4.2 it follows that if each closed subset of V S is a cone provided that
n
it contains the convex hull and intersection of any pair of its elements as well as
the dilation αx, with α≥0, of its every member x.
4.4. The proof of Theorem 4.1 shows that a positively homogeneous continuous
function f on N is the support function of a member of π↑(Ξ) if and only if
n R n
f(x ) ≥ f(y) provided that S (x ) ≥ S (y) for all ξ ∈ Ξ. Observe that we
k ξ k ξ
kP=1 kP=1
may restrict the range of the index to n = 1 only on condition that the balls S
ξ
are dilations of one another. Indeed, in this event the polar π↑(Ξ) is the weakly*
closed conic hull of two-points relations and so the functions of the form x 7→
αS (x)∧βS (x) turn out sublinear for positive α and β.
ξ1 ξ2
5. The Case of Meeting Gauges
We now address some properties of gauges which are tied with intersection.
This operation involves some peculiarities since the intersection of balls differs in
general from the pointwise infimum of their support functions. However, the idea
of decomposition applies partially to this case.
5.1. Theorem. Let H be a cone in V S and H = π (H). Assume given a
N ↓
nonzero vector y in N such that
R
S
S :=
y
S(y)
_
S∈H;S6={0}
is absorbing. Take x ,...,x in N. The inequality
1 n
R
n
S(x )≥S(y)
k
kX=1
holds for every gauge S ∈H if and only if there are vectors z ,...,z in N such
1 n
n R
that z =y and, moreover,S(x )≥S(z ) for all S ∈H.
k k k
kP=1
Proof. ⇐=: Since S is a gauge, the support function of S is a sublinear
functional and
n n n
S(x )≥ S(z )≥S z =S(y).
k k k
(cid:18) (cid:19)
Xk=1 Xk=1 kX=1
=⇒: For simplicity we restrict exposition to the case when S is absorbing for
y
every nonzero y ∈ N. Put
R
K := sup |x|.
x∈S◦
y
UPPER SEMILATTICES OF FINITE-DIMENSIONAL GAUGES 9
By hypotheses, K <+∞. We further put
U := (ν ,ν )∈C′(S )×C′(S )|ν ≥0,ν ≥0;
1 2 N−1 N−1 1 2
n
kν k∨kν k≤K; (l,·)d(ν +ν )=(l,y) (l ∈ N) ;
1 2 1 2
Z R o
SN−1
U :=U +H∗×H∗;
n
e
µ :=|x |ε ; µ := |x |ε .
1 1 x1/|x1| 2 k xk/|xk|
kX=2
As usual, we agree that the symbol |0|ε 0 stands for the zero vector.
0/|0|
Assume that the pair (µ ,µ ) does not belong to U. Since U is a weakly*
1 2
compact convex set; therefore, U is weakly* closed and convex. By the Separation
e
Theorem there are nonzero functions S′ and S′ in H such that
e 1 2
(5.1.1) µ (S′)+µ (S′)<ν (S′)+ν (S′)
1 1 2 2 1 1 2 2
for all (ν ,ν )∈U. Put
1 2
S′ S′
S := 1 ; S := 2 .
1 S′ ∧S′(y) 2 S′ ∧S′(y)
1 2 1 2
Note that S ,S ∈H. Consequently, the meet S ∧S belongs to H. Moreover,
1 2 1 2
S′ ∧S′
kykS1◦∨S2◦ =(S1◦∨S2◦)◦(y)=S1∧S2(y)= S′ 1∧S′(2y)(y)=1.
1 2
Since S ∧S ⊃S ; therefore, S◦∨S◦ ⊂S◦. In particular,
1 2 y 1 2 y
(5.1.2) sup |x|≤K
x∈S◦∨S◦
1 2
Let V be a face of S◦ ∨S◦ that contains y; i. e., the intersection of S◦∨S◦ with
1 2 1 2
some supporting hyperplane to S◦∨S◦ at y. Denote by ext(V) the set of extreme
1 2
pointsofV. BytheChoquetTheoremthereisaprobabilitymeasureν withsupport
ext(V) andbarycentery. PutV :=ext(V)∩S◦ and V :=ext(V)\V . The set V
1 1 2 1 2
lies in S◦. Let ν :=ν| and ν :=ν| . Then ν =ν +ν .
2 1 V1 2 V2 1 2
We will treat a continuous function f onS as the restrictionto S of the
N−1 N−1
unique positively homogeneous namesake function on N and put
R
ν :f 7→ fdν ;
1 1
Z
V1
ν :f 7→ fdν (f ∈C(S ));
2 2 N−1
Z
V2
ν :=ν +ν .
1 2
Using (5.1.2) and the estimate ν ( ) ≤ ν( ) = 1, with the identically one func-
1
1 1 1
tion; we see that
kν k=ν ( )= |·|dν ≤ sup |x|<K.
1 1 1
1 Z x∈S◦∨S◦
1 2
V1
10 S.S.KUTATELADZE
By analogy kν k≤K. Moreover,
2
ν(l)= (l,·)dν + (l,·)dν = (l,·)dν =(l,y)
1 2
Z Z Z
V1 V2 ext(V)
for all l∈ N. Hence, (ν ,ν ) belongs to U and
1 2
R
ν (S )+ν (S )= S dν + S dν
1 1 2 2 1 1 2 2
Z Z
V1 V2
= k·kS◦dν1+ k·kS◦dν2 =ν( )=1=S1∧S2(y).
Z 1 Z 2 1
V1 V2
By (5.1.1)
p
S ∧S (x )≤µ (S )+µ (S )<ν (S )+ν (S )
1 2 k 1 1 2 2 1 1 2 2
Xk=1
p
=S ∧S (y)≤ S ∧S (x ).
1 2 1 2 k
Xk=1
We arrive at a contradiction, which means that (µ ,µ ) lies in U; i. e. there are
1 2
measures ν , ν such that µ −ν ∈H∗, µ −ν ∈H∗, and (ν , ν )∈U. Consider
1 2 1 1 2 2 1 e2
the representing points
u :z 7→ν (z); u :z 7→ν (z) (z ∈ N).
1 1 2 2
R
Then u +u =y, and for S ∈H we have
1 2
µ (S)≥ν (S)≥S(u ); µ (S)≥ν (S)≥S(u ).
1 1 1 2 2 2
Proceed by induction and apply the above process to the measure µ and the
2
nonzero point u (it is exactly the place where we invoke the simplification of the
2
beginning of the proof). We thus come to what was desired. In case u = 0, the
2
soughtdecompositionmaybecomposedofthecopiesofthezerovectors. Theproof
is complete.
By way of illustration of Theorem 5.1 we will provide a description for π(Ξ).
5.2. Theorem. Let H be a cone in V and H =π (H). Assume that
N ↓
S
S :=
y
S(y)
^
S∈H;S6={0}
is absorbing for every nonzero y ∈ N. Then π↑(H) is closed with respect to ∧.
Moreover,and a nonzero S in V bRelongs to π↑(H) if and only if
N
S
0
(5.2.1) S = S(x)
S (x)
x^6=0 S0_∈H 0
Proof. We havealreadydemonstratedthateachS ∈π↑(H) may be written as
in(5.2.1)(cp.(4.2.2)). AssumeinturnthatShastheshape(5.2.1). ByTheorem4.1
we have to validate the implication
n n
S (x )≥S (y) for all S ∈H =⇒ S(x )≥S(y).
0 k 0 0 k
Xk=1 kX=1
