Upper semilattices of finite-dimensional gauges Текст научной статьи по специальности «Математика»

Владикавказский математический журнал октябрь-декабрь, 2006, Том 8, Выпуск 4

UDC 513.88

UPPER SEMILATTICES OF FINITE-DIMENSIONAL GAUGES

S. S. Kutateladze

In Memory of Alex Rubinov (1940-2006)

This is a brief overview of some applications of the ideas of abstract convexity to the upper semilattices of gauges in finite dimensions.

1. Introduction

Duality in convexity is a simile of reversal in positivity. The ghosts of this similarity underlay the research on abstract convexity we were engrossed in with Alex Rubinov in the early 1970s. Our efforts led to the survey [1] and its expansion in the namesake book [2]. We always cherished a hope to revisit this area and shed light on a few obscurities. However, the fate was against us.

Inspecting the archive of our drafts of these years, I encountered several items on the cones of Minkowski functionals or, equivalently, gauges. The results on the Minkowski duality in finite dimensions are practically unavailable in full form, whereas they rest on the technique that is still uncommon and unpopular but definitely profitable. The theorems on gauges appeared mostly in some mimeographed local sources that had disappeared two decades ago. We hoped and planned to expatiate on these matters when time will come.

Alex Rubinov was my friend up to his terminal day. He shared his inspiration and impetus with me. So does and will do his memory...

An abstract convex function is the upper envelope of a family of simple functions [1-3]. The cone of abstract convex elements is an upper semilattice. We describe the bipolar of such a semilattice through majorization generated by its polar. Polyhedral approximation simplifies the generators of the polar in finite dimensions to discrete measures. Decomposition reduces the matter to Jensen-type 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 majorization and decomposition in the spaces of continuous functions. Section 2 addresses the space of convex sets in finite dimensions and the influence of polyhedral approximation on the structure of dual cones. Section 3 illustrates the use of linear programming for revealing continuous linear selections over convex figures. Section 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.

© 2006 Kutateladze S. S.

2. 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 Sn-i as follows:

2.1. A measure j (linearly) majorizes or dominates a measure v provided that to each decomposition of Sn-i into finitely many disjoint Borel sets U\,, Um there are measures j1,..., jm with sum j such that every difference jk — v\uk annihilates all restrictions to Sn-i of linear functionals over HN. In symbols, we write j ^ Rnv.

Reshetnyak proved that

J pdj ^ J pdv

SN-1 SN-1

for every sublinear functional p on HN if j ^ Rn v. This gave an important trick for generating positive linear functionals over various classes of convex surfaces and functions.

2.2. A similar idea was suggested by Loomis [5] in 1962 within Choquet theory. A measure j affinely majorizes a measure v, both given on a compact convex subset Q of a locally convex space X, provided that to each decomposition of v into finitely many summands vi,..., vm there are measures ji,..., jm with j such that every difference jk — vk annihilates all restrictions to Q of the affine functions over X. In symbols, j ^ Aff(Q) v. Many applications of affine majorization are set forth in [6].

Cartier, Fell, and Meyer proved in [7] that

J fdj ^ J fdv

Q Q

for every continuous convex function f on Q if and only if j ^ Aff(Q)v.

An analogous necessity part for linear majorization was published in [8]. In applications we use a more detailed version of majorization [9]:

2.3. Decomposition Theorem. Assume that Hi,..., Hn are cones in a vector lattice X. Assume further that f and g are positive linear functionals on X. The inequality

f (hi V ■ ■ ■ V hn) > g(hi V ■ ■ ■ V hn)

holds for all hk £ Hk (k := 1,... ,n) if and only if to each decomposition of g into a sum of n positive terms g = gi + ■ ■ ■ + gN there is a decomposition of f into a sum of n positive terms f = fi +-+ fn such that

fk (hk) ^ gk (hk) (hk £ Hk; k :=1,..., n).

3. The Space of Convex Figures

We will proceed in the Euclidean space HN.

3.1. 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 HN with its support function S(z) := sup{(x, z) \ x £ S} for z £ HN. Considering the members of HN as singletons, we assume that HN lies in the set VN of all compact convex subsets of HN.

3.2. The Minkowski duality makes VN into a cone in the space C(Sn-i) of continuous functions on the Euclidean unit sphere Sn-i, the boundary of the unit ball Zn. This yields

is the so-called Minkowski structure on VN. Addition of the support functions of convex figures amounts to taking their algebraic sum, also called the Minkowski addition. It is worth observing that the linear span [Vn] of VN is dense in C(Sn-i), bears a natural structure of a vector lattice and is usually referred to as the space of convex sets. The study of this space stems from the pioneering breakthrough of Alexandrov [10] in 1937 and the further insights of Radstrom [11] and Hormander [12].

3.3. A gauge p is a positive sublinear functional on a real vector space X viewed as the

Minkowski functional of the conic segment Sp := {p ^ 1} := {x £ X \ p(x) ^ 1}. The latter

is also referred to as a gauge or caliber. A gauge p is a norm provided that its ball Sp is

symmetric and absorbing. Recall that the subdifferential or support set dp of p is the dual ball or polar of Sp. The polar of a ball S is denoted by S° and the dual norm of || ■ ||s is || ■ ||s°. The “donkey bridge” of functional analysis consists in the duality rules:

ll-lls = S°(-), II-IIs- = S (■).

We will restrict exposition to the norms and balls of HN by way of tradition.

3.4. Approximation Lemma. If H is a subcone of VN then the signed measures with finite support are sequentially weakly* closed in the dual cone H *.

< Let j £ H*. The mappings

z ^ j+(z); z ^ j_(z),

with z £ HN, are linear functionals on HN. Therefore, there are u,v £ HN such that j+(z) = (u,z) and j_(z) = (v,z). Put

Ji; = j+ + mes + \uk-«/M;

J2 := j- + mes + |v|e_v/|v|;

ji : ji + \v\^_v/|v|; j2 : j2 + \u\^_u/|u|.

As usual, ez is the Dirac measure at z £ HN, while \ ■ \ is the Euclidean norm on HN, and

mes is the Lebesgue measure on Sn_i: i. e. the surface area function of the Euclidean ball Zn := {x £ HN \ \x\ ^ 1}. Note that j = ji — j2. Moreover, the measures Ji and J2 are nondegenerate and translation-invariant. Indeed, check that so is ji. This signed measure is clearly positive and not supported by any great hypersphere. We are left with validating translation-invariance. If k := 1,... ,N then

J ej dji = J ej dj+ + J ej dj(zN) — (u,ek) = (u,ek) — (u,ek) = 0.

Sn- 1 sn-1 sn-1

Consider a convex figure x whose surface area function j(x) equals ji. The existence of this

figure is guaranteed by the celebrated Alexandrov Theorem [10, p. 108].

Let (tm.) be a sequence of polyhedra including x and converging to x in the Hausdorff metric on [Vn]which is induced by the Chebyshev norm on C(Sn_i). Then the measures jm = j(Xm) converge weakly* to ji and jm ^ Rn ji. Indeed, given a convex figure 3, we have

J 3 djm = j 3 dj(xm)= nV (h Xm,..., Xm) > nV (3, x,..., x) = j 3 dj(x)= J 3 dji

SN-1 SN-1 SN-1 SN-1

by the inclusion monotonicity of the mixed volume V(■,..., ■) in every argument.. By analogy, there is a sequence (jm), converging weakly* to j2 and such that jri ^ Hn j2. Putting

jm : jm + \v\^_v/|v|; jm : jm + \u\^_u/|u|,

we see that jm — jm converges weakly* to j. The proof is complete. >

4. Labels and Decompositions

The Approximation Lemma allows us to reduce consideration to signed measures with finite support. These measures decompose easily. We will exhibit a typical application.

4.1. A family (ji,..., jn) of regular Borel measures on the sphere Sn_i is a labeling on RN provided that (ji(x), ..., jn(x)) £ x for all x £ Vn. The vector (ji (x),..., jn(x)) is a label of x.

4.2. Proposition. A family (ji,..., jn) is a labeling on RN if and only if

n

£x — ^^ xk jk £ VN. k=i

for all x £ Sn_i-

< The Minkowski duality is an isomorphism of the relevant structures. Hence, the definition of labeling can be rephrased as follows:

n

^2xkjk(x) ^ x(x) (x £ Rn, x £ Vn). >

k=1

4.3. Using linear majorization for describing VN, we can write down some criteria for labeling in terms of decompositions. For simplicity, we will argue in the planar case.

Consider the conditions:

(++) £(A1,A2)+ Ai jl + A2 j2 > Ai j++ A2 j+;

R2

(+ ) e(A1,_A2)+ Ai jl + A2 j+ > Ai j+ + A2 j2 ;

(-----^ e(_A1,A2)+ Ai j++ A2 j2 ^ Ai jl + A2 j+;

(-----) e(_A1,_A2) + Ai j++ A2 j+ > Ai jl + A2 j2 ;

with (Ai, A2) £ Si H R+. Clearly, the requirement of 4.1 amounts to the four conditions simultaneously. By way of example, we will elaborate the relevant criterion only in the case of (+ ).

4.4. Proposition. For (+—) to hold it is necessary and sufficient that to all (Ai, A2) in Si H R + and all decompositions {(j+)i,..., (j+)m} of j+ and al decompositions {(j_)i,..., (j_)m} of j_ there exist a decomposition {(j_)i,..., (j_)m} of j_,

a decomposition {(j+)i,..., (j+)m} of j+, and reals ai,..., am that make compatible the simultaneous inequalities:

ai ^ 0;...; am ^ 0; ai + ... + am = 1;

Ai (x(M-)k — x(M+)k + ak ei ) = A2 (x(M-)k — x(M+)k + ak e2) (k -1,..., m),

where x^ is the representing point of j; i. e., j(u) = (u, x^) for all u £ H2.

< ^=: Let (Ai, A2) £ Si H R+ and let {vi,...,vm} be an arbitrary decomposition of Ai j++ A2 j_. By the Riesz Decomposition Lemma there are a decomposition {(j+)i,..., (j+)m} of j+ and a decomposition {(j_)i,..., (j_)m} of j_ such that Ai (j+)k + A2 (j_)k = vk. Find some parameters satisfying the simultaneous inequalities and put

jk : = Ai (ji )k + a2 (jt)k + ak£(a1,-A2).

Clearly, jk ^ 0 and, moreover,

y^yjk =Ai j! + A2 j+ + £(A1,-A2).

k=1

Furthermore,

xVk — xVk =Ai x(M-)k + A2 x(M+)k + ak Ai ei — ak A2 e2— Ai x(M+)k — A2 x(M2-)k = °

and so jk — vk belongs to the polar of H2 in C (Si).

=^: Assume (+—) valid.

Given decompositions {(j+)i,..., (j+)m} and {(j_)i,..., (j_)m} there is a decomposition {vi,..., v2m} of £(A1j-A2)+ Ai j_+ A2 j+ such that

xVk = x(M+)k ; xVm+k = x(M2-)k (k :=1,...,m).

We are left with appealing to the Riesz Decomposition Lemma and representing the decomposition {vi,..., v2m} through the corresponding decompositions of £(A1;_A2), Ai j_, and A2 j+. The proof is complete. >

4.5. If it is possible to chose decompositions in 3.4 independently of (Ai, A2), then 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

ji := \ j+\eJu+/|Ju+| — \j \£^-/|^-|;

I +1 l—l

j2 := \v \ev+/|v+| — \v \ev-/|v-1,

with j+, j_, v +, and v_ some points on the plane. The sufficient condition we have just

suggested paraphrases as follows:

ak, $k, ak, bk, ^k, ck ^ 0; ak + ak = 1; /3k + bk = 1; Yk + Ck = 1 (k :=1,..., 4);

j+ = aij + Yiei; 3i v + Yie2 = 0;

biv _ + cie2; ai j_ + ciei = 0;

+

H

— — ,,+

a2j+ — Y2ei; 32 v + Y2e2 = 0;

v+ = b2v _ + C2e2; a2 j+ — C2ei = 0;

H

= a3j+ — Y3ei; 33 v + — Y3e2 = 0;

H

v = b3v+ — C3e2; a3 j+ — C3ei = 0;

a4 j_ + Y4ei; 34 v + — Y4e2 = 0;

+

v = b4v+ — c4 e2; a4 j + Y4ei = 0.

The solution of the last system is given by the parameters:

1

ak = bk = 0; 3k = ak = 1; Yk = Ck = ^ (k := 1,..., 4).

Moreover,

H

+

2

e1;

+

e2; j

— 2 ei;

— 2 e2.

Therefore, the simplest labeling of x is the point 2(x(ei) — x(—ei),x(e2) — x(—e2)). It 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.

1

1

1

v

5. 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 VN. To simplify notation we will discuss only balls, denoting the set of balls in VN by VSn . It is convenient formally to add the apex to VSn . If S £ VNS differs from the origin then we use the symbol | ■ |s not only for the gauge of S but also for the operator norm corresponding to S in the endomorphism space L (HN) o^HN. In other words,

||x||s := inf {a > 0 \ x/a £ S} (x £ HN);

||A||s := sup{|Ax|s \ x £ S} (A £ L(RN)).

Recall that

S ° = {x ^N \\(x,y)\ < 1(y £ S)},

where (■, ■) is the standard inner product of RN.

Observe that VNS is a lattice and simultaneously a cone. However, VNS is not closed in VN. This circumstance notwithstanding, given a family (S£)^es in VNS, sometimes we may soundly speak of the upper hull n^(S), lower hull n^(S), and hull n(S) of this family, implying the least closed cones that lie in VNS, include S£ for all £ £ S, and are closed under the join, the meet, and both operations in the lattice of convex figures VN. An example is provided by

any instance of nondegenerate family. The latter is by definition any family of nonzero sets

(S£)£gs such that,

sup ||A||Sf < (A £ L^N)).

£ss

Indeed, put

A(S) := {A £ L(RN) \ AS£ C S£ (£ £ S)},

and let M(S) be the set of the symmetric elements of VN such that AS C S for all A £ A(S). Since (S£)£gs is nondegenerate, all members of M(S) but the zero singleton are absorbing. Moreover, M(S) is clearly a closed sublattice of VN.

We will need the helpful property of a nondegenerate family: If y £ RN differs from the zero of RN then

s = A

: /4 S£ (y)

is absorbing. Indeed, given z £ RN we infer that

sup S£(y)S°(z) = sup ||y|s2 l|z||se = sup ||y 0 z|se < +TO,

£SS £eS £eS

where y0z : x ^ (y, x)z for all x £ RN. Hence, the polar of Sy is compact, which implies that

Sy is absorbing. Without further specification, we will address only nondegenerate families of balls in the sequel.

5.1. Theorem. A gauge S belongs to n^(S) if and only if

s * w s£

n ^ V n

E |xklls? £es E |xk||s?

k=1 k=1

for any collection of the vectors xi,... ,xp £ RN that are not all zero simultaneously.

< It is obvious that n^(S) is the closure of the upper semilattice of all H-convex functions

with H the conic hull of the family (S£)£es. The polar of n^(S) may be approximated with finitely supported signed measures by the Approximation Lemma. Using the Bipolar Theorem, we see that S £ (S) if and only if ^n=i S(xk) ^ S(y) whenever y,xi,... ,xn £ RN satisfy

En=i S£(xk) ^ S£(y) for all £ £ S. By duality, S £ n^(S) if and only if

nn

ADl*kIs?s° cJ2\\xkIs-s°.

£SS k=i k=i

Taking polars, we complete the proof of the theorem. >

5.2. Corollary. A nonzero gauge S belongs to n^ (S) if and only if

n s

(4.2.1) s = A ES(xk)V ~ £-----------•

(x1,...,xn) k=i £e^ S£ (xk)

k=1

where the intersection ranges over all nonzero tuples (xi,..., xn) £ RN.

< Clearly, (4.2.1) guarantees the inclusion of 4.1 and so S £ n^ (S). The last containment in turn implies the simple representation:

s

(4Z2) s = As(x) V sfc.

x=0 £SS £V ’

Indeed, denote by S the right-hand side of (4.2.2). By 4.1, S * S'. If z £ Hn then

S(z)=(s (x) W s^)(z) *s (z)( W ^(z)=s (z) W =s (z).

By the Minkowski duality S * S. Denote by S the right-hand side of (4.2.1). Since S * S * S * S; therefore, S = S and we are done. >

5.3. From 4.2 it follows that if each closed subset of VnS is a cone provided that it contains the convex hull and intersection of any pair of its elements as well as the dilation ax, with a ^ 0, of its every member x.

5.4. The proof of Theorem 4.1 shows that a positively homogeneous continuous function f onRN is the support function of a member of n^(S) if and only if En=i f (xk) ^ f (y) provided that En=i S£(xk) ^ S£(y) for all £ £ S. Observe that we may restrict the range of the index to n = 1 only on condition that the balls S£ are dilations of one another. Indeed, in this event the polar n (S) is the weakly* closed conic hull of two-points relations and so the functions of the form x ^ aS£1 (x) A 3S£2 (x) turn out sublinear for positive a and 3.

6. 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.

6.1. Theorem. Let H be a cone in VNS and H = n^(H). Assume given a nonzero vector y inM,N such that

V —

seXfo} S(y)

Sy : =

is absorbing. Take xi,... ,xn inUN. The inequality

Y,S(xk) ^ S(y)

k=i

holds for every gauge S £ H if and only if there are vectors zi,... ,zn in RN such that En=i zk = y and, moreover, S(xk) ^ S(zk) for all S £ H.

< ^=: Since S is a gauge, the support function of S is a sublinear functional and

ES (xk) ^ S (zk) ^ S( ][> ) = S(y).

k=i

k=i

k=i

>: For simplicity we restrict exposition to the case when Sy is absorbing for every nonzero

y £TR,n. Put

K := sup \x\.

xesy

By hypotheses, K < +ro. We further put

U := { (vi^ v2) £ C /(SN—i) X C' (SN—i) \ vi ^ 0^ v2 ^ 0;

V|v2|| * K; J (l, ■)d(vi + v2) = (l,y) (l £M,N)};

SN-1

U := U + H* X H*;

n

ji : \xi\^x1/|x1|; j2 : ^ ' \xk\^xk/|xk |.

k=2

As usual, we agree that the symbol \0\£q/|o|0 stands for the zero vector.

Assume that the pair (ji, j2) does not belong to U. Since U is a weakly* compact convex set; therefore, U' is weakly* closed and convex. By the Separation Theorem there are nonzero

functions Si and S2 in H such that

(5.1.1) ji(Si)+ j2(S2) <vi(Si) + v2(S2)

for all (vi, v2) £ U. Put

Of Of

si . c s2

Si A S2 (y)’ ' Si A S2 (y)'

Note that Si, S2 £ H. Consequently, the meet Si A S2 belongs to H. Moreover,

sf a Sf

|y|s?vs2 = (S° V S°)°(y) = Si A S2(y) = si A s2 (2y) (y) = 1.

Since Si A S2 D Sy; therefore, S° V S° C S°. In particular,

(5.1.2) sup \x\ * K

xes-vs|

Let V be a face of S° V S° that contains y; i. e., the intersection of S° V S° with some supporting hyperplane to S° V S° at y. Denote by ext(V) the set of extreme points of V. By the Choquet Theorem there is a probability measure v with support ext(V) and barycenter y. Put Vi := ext(V) H S° and V2 := ext(V) \ Vi. The set V2 lies in S°. Let vi := v\y1 and v2 := v\V2. Then v = vi + v2.

We will treat a continuous function f on Sn_i as the restriction to Sn_i of the unique positively homogeneous namesake function on HN and put

vi : f ^ J fdv i;

V1

v2 : f fdv2 (f £ C(Sn_i));

V2

v := vi + v2.

Using (5.1.2) and the estimate vi^) * v^) = 1, with 1 the identically one function; we see that

|vi| = vi(l) = \ ■ \dvi * sup \x\ < K.

J xeS?VS2

V1 1 2

By analogy 11 v21 * K. Moreover,

v(l) = J(l, ■)dvi + f(l, ■)dv2 = J (l, ■)dv =(l,y)

V1 V2 ext(V)

for all l £ Rn. Hence, (vi,v2) belongs to U and

vi(Si) + v2(S2) = J Sidvi + J S2dv2

V1 V2

|s?dvi + J || ■ ||s?dv2 = v(^) = 1 = Si A S2(y).

V1 V2

By (5.1.1)

j^Si A S2(xk) * ji(Si) + j2(S2) < vi(Si) + v2(S2)

k=1

P

= Si A S2(y) *£ Si A S2(xk).

k=1

We arrive at a contradiction, which means that (ji, j2) lies in U; i. e. there are measures vi, v2 such that ji — vi £ H*, j2 — v2 £ H*, and (vi, v2) £ U. Consider the representing points

ui : z ^ vi(z); u2 : z ^ v2(z) (z £ RN).

Then ui + u2 = y, and for S £ H we have

ji(S) ^ vi(S) ^ S(ui); j2(S) ^ v2(S) ^ S(u2).

Proceed by induction and apply the above process to the measure j2 and the nonzero point u2 (it is exactly the place where we invoke the simplification of the beginning of the proof). We thus come to what was desired. In case u2 = 0, the sought decomposition may be composed of the copies of the zero vectors. The proof is complete. >

By way of illustration of Theorem 5.1 we will provide a description for n(S).

6.2. Theorem. Let H be a cone in VN and H = n^(H). Assume that

V = A S

y ' ' ' S(y)

SsH;S={0}

is absorbing for every nonzero y £ RN. Then (H) is closed with respect to A. Moreover, and a nonzero S in VN belongs to n (H) if and only if

<5-2-1> s = A s(x) V sSox)

x=0 SqGH 0( )

< We have already demonstrated that each S £ n^(H) may be written as in (5.2.1) (cp. (4.2.2)). Assume in turn that S has the shape (5.2.1). By Theorem 4.1 we have to validate the implication

nn

]T So(xk) ^ So(y) for all So £ H =^ ]T S(xk) ^ S(y).

k=1 k=1

Since H = (H), by Theorem 4.1 there are vectors zi,... ,zn such that

n

zk = y;

k=1

So (xk) ^ So(zk) (So £ H)

Since S is represented as (5.2.1), S(xk) ^ S(zk). Hence,

n n / n \

Y^S(xk) S(zk) ^ ^^zJ = S(y).

k=1 k=1 k=1

Thus, S £n j (H).

We are left with checking that (H) is closed under A. By above, S £ (H) if and only

if S(x) ^ S(y) for all x,y £ HN satisfying So(x) ^ So(y) for all So £ H.

So, take Si, S2 £ n(H) and assume that So(x) ^ So(y) for all So £ H.

We are to compute Si A S2(x). Arguing as in Theorem 5.1 and replacing the reference to the Choquet Theorem to the Caratheodory Theorem, find vectors xi,...,xn such that kn=1 xk = x and

t n

Si A S2(x) = J2 Si (xk)+ ^ S2(xk).

k=1 k=t+1

If So £ H then

nP

(xk) ^ SoMTxA = So(x) ^ So(y).

k=1 k=1

Hence, by Theorem 5.1 there are vectors zi,... ,zn £ HN such that ^n=i zk = y and So(xk) ^ So(zk) for all So £ H and k := 1,...,n. Thus, Si(xk) ^ Si(zk) and S2(xk) ^ S2(zk). Consequently,

t n t n

Si A S2(x) = J] Si (xk)+ ]T S2(xk) ^ Si (zk)+ ]T S2(zk)

k=1 k=t+1 k=1 k=t+1

n / n \

^ Si A S2(zk) ^ Si A S2(]T zk\ = Si A S2 (y).

k=1 k=1

Therefore, Si A S2 belongs to n^(H), which completes the proof. >

6.3. Corollary. Let (S£)£S2 be a nondegenerate family of balls. Then

n(S) = nT(ni(S)).

In this event a nonzero gauge S belongs to n(S) if and only if

So

S = A S(x) v

So(x)' x=o SoSn|(s)

< Obviously, nT(n^(S)) lies in n(S). Note now that

s = A So -) A So

Soen(S);So={o} So(y) SoSM(S);So={o} So(y)

The family M(S) is nondegenerate since so is (S£)£es. Hence, Sy is absorbing. By Theorem 5.2 n(n^(S)) is closed under A, thus serving as a superset of n(S). >

6.4. In study of the properties of gauges with are related to intersection, we have actually used the accompanying representation

(5.4.1) [ Si A S2dj = inf ( [ Sidji + [ S2dj2),

J M1+M2 > J J J

Sn-1 R Sn-1 Sn-1

which generalizes the standard formula for the infimal convolution □, a routine operation of convex analysis:

Si A S2 = Si^ S2.

It is an easy matter to see the lattice-theoretic provenance of (5.4.1). Some slightly annoying subtlety of the general case which was obviated by finite dimensionality is connected with the fact the infimum of abstract convex elements in the lattice of these elements is just a partial superlinear operator.

Acknowledgement. The main results of this article stem from our joint work with Alex Rubinov by the mid 1970s. I gratefully emphasize his creative contribution to all areas of abstract convexity we had been exploring those happy years.

References

1. Kutateladze S. S., Rubinov A. M. Minkowski Duality and Its Applications // Russian Math. Surveys.— 1972.—V. 27, № 3.—P. 137-191.

2. Kutateladze S. S., Rubinov A. M. Minkowski Duality and Its Applications.—Novosibirsk: Nauka Publishers, 1976.

3. Rubinov A. M. Abstract Convexity and Global Optimization.—Dordrecht: Kluwer Academic Publishers, 2000.

4. Reshetnyak Yu. G. On the Length and Swerve of a Curve and the Area of a Surface (Ph. D. Thesis). Leningrad State University [in Russian].—1954.

5. Loomis L. Unique direct integral decomposition on convex sets // Amer. Math. J.—1962.—V. 84, № 3.— P. 509-526.

6. Marshall A. W., Olkin I. Inequalities: Theory of Majorization and Its Applications.—New York: Academic Press, 1979.

7. Cartier P., Fell J. M., Meyer P. A. Comparaison des mesures poertees par un ensemble convexe compact // Bull. Soc. Math. France.—1964.—V. 94.—P. 435-445. [in French].

8. Kutateladze S. S. Positive Minkowski-linear functionals over convex surfaces // Soviet Math. Dokl.— 1970.—V. 11, № 3.—P. 767-769.

9. Kutateladze S. S. Choquet boundaries in K-spaces // Russian Math. Surv.—1975.—V. 30, № 4.—P. 115155.

10. Alexandrov A. D. Selected Scientific Papers.—London etc.: Gordon and Breach, 1996.

11. Radstrom H. An embedding theorem for spaces of convex sets // Proc. Amer. Math. Soc.—1952.—V. 3, № 1.—P. 165-169.

12. Hormander L. Sur la fonction d’appui des ensembles convexes dans une espace lokalement convexe // Arkiv for Math.—1955.—V. 3, № 2.—P. 180-186. [in French].

Received by the editors December 31, 2006.

Prof. Kutateladze Semen Samsonovich Novosibirsk, Sobolev Institute of Mathematics E-mail: sskut@math.nsc.ru