Journal of Convex Analysis Volume 2 (1995), No.1/2, 291–307

Antihomogeneous Conjugacy Operators in Convex Analysis

Alexander Rubinov

Ben-Gurion University of the Negev, Faculty of Natural Science, Department of Mathematics and Computer Science, Beer-Sheva 84105, Israel.

Dedicated to R. T. Rockafellar on his 60th Birthday

Received 30 June 1994 Revised manuscript received 11 October 1994

An axiomatic description of antihomogeneous conjugacy operators is suggested and some properties of these operators are established. Moreover several examples are given.

Keywords : conic complete lattice, antihomogeneous conjugacy operator, calculus, operator of polarity.

1. Introduction

One of the main methods of Convex Analysis is the application of conjugacy operators which transfer a given convex object (set, function, multivalued mapping and so on ) to a conjugate object. Several conjugacy operators are used in Convex Analysis, for instance the operator of polarity which maps a convex closed set containing the origin to its polar set, the operator which transfers the same set to its Minkowski gauge, the operator which maps a convex lower semicontinuous (l.s.c.) function in its conjugate function (in the sense of Convex Analysis). Various kinds of conjugacy operators are described in the famous book by Rockafellar [14]. Conjugacy operators are used also in the theory of quasiconvex functions (see for instance [5, 11, 13, 17]). It is most convenient to study conjugacy operators within framework of lattice theory as dual isomorphism between lattices. This approach was suggested the famous Russian mathematician G.P.Akilov in the 1970’s. Unfortunately the important work of G.P.Akilov was not published. Varios classes of conjugacy operators and varios aspects of the theory of conjugacy operators are studied in framework of lattice theory by I. Singer, J.-E. Martinez-Legaz (see for example [20, 12]) and others. There is a survey of results in this direction in the paper [10] by J.-E. Martinez-Legaz. On the other hand a vector version of some conjugacy operators of Convex Analysis (with substitute functionals for operators and straight line for vector lattices) was studied by A.G.Kusraev and S.S.Kutateladze (see for example [6, 7]). Conjugacy operators which are studied in Convex Analysis and related areas are defined as a rule on a lattice with multiplication by positive scalars, and these operators are

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292 A. Rubinov / Antihomogeneous conjugacy operators in convex analysis

antihomogeneous (i.e. homogeneous of degree (-1)). We are primarily concerned in this paper with conjugacy operators possessing this antihomogeneuos property. The aim of this paper is to give an axiomatic description of antihomogeneous conjugacy operators and to study connections between these operators and algebraic operations (calculus). Our viewpoint sheds some new light on the calculus and allows us to obtain some results by the simplest way.

Examples are part and parcel of the paper.

In Section 2 we introduce so-called c^{2}-lattices which are natural domains of antihomogeneous conjugacy operators. In Section 3 antihomogeneous conjugacy operators are introduced. Sections 4 and 5 are devoted to the study of some algebraic operations and calculus. Finally in Section 6 the operator of polarity is studied.

2. c^{2} -lattices.

The natural domain of an antihomogeneous conjugacy operator is a complete lattice with multiplication by a positive scalar.

Definition 2.1. A set U is called conic complete lattice (c^{2}-lattice) if

i) U is a conic set, i.e. multiplication by positive scalar is defined on U with properties

as follows:

1u = u; λ(μu) = (λμ)u ∀u ∈ U, ∀λ, μ > 0.

ii) U is a complete lattice i.e. U is an ordered set and there exist supremum and infimum

for an arbitrary subset of the set U. Let us give some examples of c^{2}-lattice. We consider three situations. Situation 1. c^{2}-lattices of nonnegative functions. Let \overline{R}_{+} = [0, +∞] be the closed ray of all nonnegative numbers together with +∞. Let us consider a set U of functions defined on a set X and acting into the ray \overline{IR}_{+} with properties as follows

1) 0 ∈ U;

2) If u ∈ U, λ > 0 then λu ∈ U where (λu)(x) = λu(x) ∀x ∈ X;

3) If A ⊂ U then \overline{sup}A ∈ U where \overline{sup}A is an upper envelope of the set A, i.e. (\overline{sup}A)(x) = sup_{f∈A} f(x) ∀x ∈ X.

We consider U as an ordered set with natural order relation : u_{1} ≥ u_{2} if and only if u_{1}(x) ≥ u_{2}(x) ∀x ∈ X. Let A ⊂ U. Clearly the function \overline{sup}A is the supremum of the set A in the ordered set U. If f is an arbitrary function defined on X and acting into \overline{IR}_{+} then a function co_{U} = \overline{sup}\{u ∈ U : u ≤ f\} is called the U-hull of the function f.

Clearly a function co_{U}(\overline{inf}A) is the infimum of the set A in the ordered set U. Here and subsequently \overline{inf}A stands for the lower envelope of the set A. We will denote \overline{sup}A and co_{U}(\overline{inf}A) as sup A and inf A accordingly. Thus U is a complete lattice. Clearly U is a c^{2}-lattice with regard to natural multiplication on a positive number and natural order relation. Let us give some concrete examples of c^{2}-lattices which are sets of functions with properties 1) - 3) above.

1. A set F (X) of all functions defined on a set X and acting into \overline{IR}_{+}.

2. Sets M_{0}(X) of all increasing functions and M^{0}(X) of all decreasing functions defined on an ordered set X and acting into \overline{IR}_{+}.

Below we will consider only functions defined on a locally convex Hausdorff space X and acting into \overline{IR}_{+}. Recall that a function f is called proper if there is x ∈ X such that f(x) ≠ +∞. Let us continue the list of examples.

3. A set P H(X) of all lower semicontinuous (l.s.c.) nonnegative positively homogeneous of degree one proper functions.

4. A set SL(X) of all nonnegative proper sublinear l.s.c. functions.

5. A set CF (X) of all nonnegative proper convex l.s.c. functions f such that f(0) = 0.

6. A set Q(X) of all nonnegative quasiconvex l.s.c. functions f with property f(0) = 0. Let us note that SL(X) = P H(X)∩CF (X). At the same time SL(X) = P H(X)∩Q(X). This was proved by J. P. Crouzeix (see, for example, [5]). Situation 2. c^{2}-lattices of H-convex functions. Let Y be a cone in a locally convex Hausdorff space X and H be a cone of functions defined on the Y and acting into IR = (−∞, +∞) . A function f defined on Y and acting into \overline{IR} = (−∞, +∞] is called H-convex (see [8]) if there is a set A ⊂ H such that f(y) = sup_{h∈A} h(y) ∀y ∈ Y . We will denote by C^{H}(Y) the set of all H-convex functions. It easy to check (see [8]) that C^{H}(Y) is a convex cone. The set C_{∗}^{H}(Y) = C^{H}(Y)∪{−∞}, where (−∞)(x) = −∞ ∀x is a complete lattice as regards the natural order relation: f_{1} ≥ f_{2} if and only if f_{1}(y) ≥ f_{2}(y) ∀y . Really if A ⊂ C^{H}(Y) then sup A = \overline{sup}A ; if A bounded from below by a function h ∈ H, i.e. h ≤ f ∀f ∈ A then inf A = co_{H} \overline{inf}A where co_{H}g = sup{h ∈ H : h ≤ g} ; otherwise inf A = −∞. There are various ways in order to introduce multiplication by a positive scalar in the lattice C_{∗}^{H}(Y). We need the following two methods:

(λ · f)(x) = λf(x)  (1)
(λ ⊗ f)(x) = \frac{1}{λ}f(λx).  (2)

Accordingly we consider two c^{2}-lattices in this case. We will write C_{1}^{H}(Y) for C_{∗}^{H}(Y) with multiplication · defined by formula (1), and C_{2}^{H}(Y) for C_{∗}^{H}(Y) with multiplication ⊗ defined by formula (2). Let us give some examples.

1. If Y = X and H consist of all affine continuous functions defined on the space X then C_{∗}^{H}(X) is the set of all convex l.s.c. functions defined on this space.

2. Let Y be a closed cone, Y − Y = X and H be the set of all nonnegative linear functions h defined on Y with the next property: there is a continuous linear function ĥ on X such that h is restriction of ĥ. A sublinear l.s.c. function p̂ defined on X is called a monotonic extension of a sublinear l.s.c. function p defined on Y if p̂(y) = p(y) ∀y ∈ Y and p̂(x_{1}) ≥ p̂(x_{2}) if x_{1}, x_{2} ∈ X and x_{1} − x_{2} ∈ Y. It can be shown (see [8]) that C^{H}(Y) is the set of all sublinear l.s.c. functions defined on Y which have a monotonic extension.

3. Let X be an Euclidean space with inner product [·, ·] and Y = X. Let us consider the cone H of all quadrics h which have a form

h(x) = α ∥ x ∥^{2} +[l, x] + β,

where α ≤ 0, l ∈ X, β is a scalar. Applying results of the papers [8, 2] it is easy to check that C^{H}(X) is the set of all l.s.c. functions defined on X and bounded from below by a function h ∈ H.

Situation 3. c^{2}- lattices of sets. Let U be a totality of subsets of a vector space X such that

1) 0 ∈ Ω for all Ω ∈ U ; X ∈ U;

2) If Ω ∈ U then λ · Ω = {λx : x ∈ Ω} ∈ U for all λ > 0 ;

3) If A ⊂ U then ⋂{Ω : Ω ∈ A} ∈ U . If Ω is a subset of the space X and 0 ∈ Ω then U- hull of Ω, denoted by co_{U}Ω is the set ⋂{Ω′ : Ω′ ∈ U, Ω′ ⊃ Ω}. Let us consider U as an ordered space: Ω_{1} ≥ Ω_{2} if and only if Ω_{1} ⊃ Ω_{2} . It is easy to check that for arbitrary subset A of the set U there exist supremum and infimum of A in the ordered set U and the next formulae hold

inf A = ⋂{Ω : Ω ∈ A}; sup A = co_{A} ⋃{Ω : Ω ∈ A}.

Clearly U is a c^{2}-lattice as regards the natural multiplication of a positive number and natural order relation. Let us give some concrete examples of c^{2}-lattices of sets with properties 1) - 3) above.

1. A set ST (X) of all star with regard to zero closed subsets of a locally convex space X. Recall that a subset Ω of the space X is called star with regard to zero if relation x ∈ Ω implies λx ∈ Ω for all λ ∈ [0, 1] .

2. A set CS(X) of all convex closed subsets of the space X such that 0 ∈ X.

3. A set Cone(X) of all closed convex cones K ⊂ X.

4. A set Lin(X) of all closed linear subspaces of the space X.

3. Antihomogeneous conjugacy operators

Definition 3.1. Let U and V be c^{2}-lattices. A mapping P : U → V is called antihomogeneous conjugacy operator if

1. The mapping P is a dual isomorphism of lattices U and V i.e. P is one-to-one correspondence and inequality P u ≥ P v is true if and only if u ≤ v .

2. P (λu) = \frac{1}{λ}P (u) for all u ∈ U and λ > 0 .

Clearly if P : U → V is an antihomogeneous conjugacy operator then there exist the inverse operator P^{−1} : V → U and P^{−1} is an antihomogeneous conjugacy operator too. Let us give some examples of antihomogeneous conjugacy operators. We use the denotations of c^{2}-lattices which describe above without reminding.

Example 3.2. Mappings P : F (X) → F (X) , P : M_{0}(X) → M^{0}(X) , P : M^{0}(X) → M_{0}(X) where P (f)(x) = \frac{1}{f(x)} are antihomogeneous conjugacy operators.

Example 3.3. Let Ω be a star with regard to zero closed set and μ_{Ω}(x) = inf{λ > 0 : x ∈ λΩ} is a Minkowski gauge of Ω. (We assume that inf ∅ = +∞). The operator P : ST (X) → P H(X) where P (Ω) = μ_{Ω} is an antihomogeneous conjugacy operator.

In examples below we will consider dual vector spaces X and L with a bilinear form [·, ·].

Example 3.4. 3. Mappings P : CS(X) → CS(L) , P : Cone(X) → Cone(L) and P : Lin(X) → Lin(L) defined by the formula P (Ω) = Ω^{∘} are antihomogeneous conjugacy operators. Here Ω^{∘} is the polar set of Ω.

Example 3.5. Let P : SL(X) → SL(L) be the operator defined by formula P (f) = f^{∘} , where f^{∘} is the polar function of f. Then P is an antihomogeneous conjugacy operator. Recall (see [14]) that

f^{∘}(l) = inf{μ > 0 : [l, x] ≤ μf(x) ∀x}.

Example 3.6. A mapping P : CF (X) → CF (X) where

P (f)(l) = inf{μ > 0 : [l, x] ≤ 1 + μf(x) ∀x}

is an antihomogeneous conjugacy operator. This operator was introduced and studied by R. T. Rockafellar [14].

Example 3.7. A mapping P : Q(X) → Q(X) where

P (f)(l) = sup_{[l,x]>1} \frac{1}{f(x)}

is an antihomogeneous conjugacy operator. This operator was introduced and studied in [17].

Example 3.8. Let X_{1} and X_{2} be locally convex Hausdorff spaces, Y ⊂ X_{1} and H ⊂ X_{2} be closed convex cones. Let us consider a function (y, h) → ⟨y, h⟩ defined on the Cartesian product Y × H with properties as follows:

⟨λy, h⟩ = ⟨y, λh⟩ = λ⟨y, h⟩ ∀λ > 0, y ∈ Y, h ∈ H;
∀y_{1}, y_{2} ∈ Y ∃h ∈ H such that ⟨y_{1}, h⟩ ≠ ⟨y_{2}, h⟩;
∀h_{1}, h_{2} ∈ H ∃y ∈ Y such that ⟨y, h_{1}⟩ ≠ ⟨y, h_{2}⟩.

This function allows us to consider an element y ∈ Y as a function defined on H and an element h ∈ H as a function defined on Y. We will write either y(h) or h(y) instead of ⟨y, h⟩ in these cases.

Let \overline{Y} be the set of all functions f defined on H which have a form f(h) = y(h) + c and let \overline{H} be the set of all functions g defined on Y such that g(y) = h(y) + c where

y ∈ Y, h ∈ H, c ∈ R . Now we consider c^{2}-lattices C\overline{H} ; 1(Y) and C\overline{Y} ; 2 (H) which were defined in the Situation 2 of the Section 2. Set

(P f)(h) = sup_{y∈Y}(⟨y, h⟩ − f(y)).  (3)

It is well known (see for example [8, 2]) that P is a dual isomorphism of the complete

lattices C\overline{H} ; 1(Y) and C\overline{Y} ; 2 (H). At the same time P (λ · f) = \frac{1}{λ} ⊗ P (f) for all λ > 0. So P is an antihomogeneous conjugacy operator.

Let us return to the dual vector spaces X and L. Let X_{1} = X, X_{2} = L , Y = X and H = L. Clearly f ∈ C\overline{H}(X) if and only if f is a l.s.c. convex function defined on the space X. In this case the function P (f) coincides with the conjugate function f^{∗} in the sense of Convex Analysis.

Example 3.9. Here we will mention antihomogeneous conjugacy operators defined on c^{2}-lattices of convex processes. Recall that a multivalued mapping a : X_{1} → X_{2} is called a convex process if a(x+y) ⊃ a(x)+a(y) and a(λx) = λa(x) ∀λ > 0. Conjugacy operators for finite dimensional convex processes were introduced and studied by R. T. Rockafellar [14, 15]. There is a complete theory of conjugate convex processes in the paper [4] by J. M. Borwein. (Some results from [4] were made more precisely in [21]. We consider two pairs X_{1}, L_{1} and X_{2}, L_{2} of dual vector spaces and a convex process a : X_{1} → X_{2}. The convex process a^{∗} : L_{2} → L_{1} is called conjugate with regards to a if we have for all g ∈ L_{2}

a^{∗}(g) = {f ∈ L_{1} : [f, x] ≥ [g, y] ∀x ∈ X, y ∈ a(x)}.

It is convenient for applications to Mathematical Economics to consider only convex processes with the following properties:

i) dom a = {x ∈ X_{1} : a(x) ≠ ∅} coincides with a given cone K_{1}. ii) If x ∈ K_{1} then a(x) is part of a given cone K_{2} and a(x) is a normal set: a(x) =

cl((a(x) − K_{2}) ∩ K_{2}). The conjugate mapping a^{∗} is defined in this case only on a cone K_{2}^{∗} by the formula

a^{∗}(g) = {f ∈ K_{1}^{∗} : [f, x] ≥ [g, y] ∀x ∈ K_{1}, y ∈ a(x)}.

Here K^{∗} is the conjugate cone for the cone K. A conjugate mapping in this form and its generalization are studied in detail by author [9, 16]. We must define various c^{2}-lattices of convex processes in order to study various kinds of conjugacy operators.

4. Calculus

The connection between an antihomogeneous conjugacy operator and multiplication by a positive number is given in the definition of an antihomogeneous conjugacy operator. It is easy to verify using the simplest properties of lattices (see for example [3]) that the following is true.

Proposition 4.1. Let P : U → V be an antihomogeneous conjugacy operator and A a subset of a c^{2}-lattice U. Then

P (sup_{u∈A} u) = inf_{u∈A} P (u); P ( inf_{u∈A} u) = sup_{u∈A} P (u).

Summation is not present in the definitions both of a c^{2}-lattice and an antihogeneous conjugacy operator, but there is a lot of c^{2}-lattices with summation and it is very interesting to establish connections between summation and the antihomogeneous conjugacy operator. At first we define multiplication by zero and +∞ . Let U be a c^{2}-lattice and u ∈ U . By definition

0 · u = inf_{0<λ<+∞} λu, +∞ · u = sup_{0<λ<+∞} λu.

Now we introduce two algebraic operations defined on the c^{2}-lattice U. If u, v ∈ U then we set

u ˆ+v = inf_{0≤α≤1} sup (\frac{1}{α}u, \frac{1}{1 − α}v) ,  (4)
u ⊕ v = sup_{0≤α≤1} inf (αu, (1 − α)v) .  (5)

Let us note then these operations were introduced and studied for c^{2}-lattice Q(X) of all nonegative quasiconvex l.s.c.functuons q such that q(0) = 0 in [17].

Proposition 4.2. Let P : U → V be an antihomogeneous conjugacy operator. Then P (u ˆ+v) = P u ⊕ P v, P (u ⊕ v) = P u ˆ+P v .

Proof. It is clear. Let u, v ∈ \overline{IR}_{+}. It is easy to check (see for example [18]) that

inf_{0≤α≤1} sup (\frac{1}{α}u, \frac{1}{1 − α}v) = u + v.  (6)

Straightforward calculation shows also that

sup_{0≤α≤1} inf (αu, (1 − α)v) = \frac{1}{\frac{1}{u} + \frac{1}{v}} .  (7)

We now apply formulae (6) and (7) in order to describe the operations ˆ+ and ⊕ defined on special c^{2}-lattices of functions. We will concider a c^{2}-lattice U of functions defined on a set X such that

i) the order relation defined on U determined by natural way: u_{1} ≥ u_{2} if and only if u_{1}(x) ≥ u_{2}(x) ∀x ∈ X;

ii) if A ⊂ U then sup A = \overline{sup}A where \overline{sup}A is an upper envelope of the set A. In the sequel U denotes the totality of all c^{2}-lattices with properties i) and ii). Clearly if U ∈ U and A ⊂ U then inf A ≤ \overline{inf}A where \overline{inf}A is the lower envelope of the set A. Let us note that c^{2}-lattices of functions which were indicated in the Section 2 (see situation 1 and situation 2 in this section) are members of the totality U.

Proposition 4.3. Let U ∈ U.

1) Suppose that the pointwise sum u_{1} + u_{2} ∈ U whenever u_{1}, u_{2} ∈ U. Then u_{1} ˆ+u_{2} = u_{1} + u_{2} .

2) Suppose that the pointwise infimum \overline{inf}(u_{1}, u_{2}) ∈ U whenever u_{1}, u_{2} ∈ U. Then u_{1} ⊕ u_{2} coinsides with the inverse sum u_{1} ˙+u_{2} where

u_{1} ˙+u_{2} = \frac{1}{\frac{1}{u_{1}} + \frac{1}{u_{2}}} .

Let us note that the function u_{1} ˙+u_{2} belongs to U in this case.

Proof. 1) Let

w_{α} = \overline{sup} (\frac{1}{α}u_{1}, \frac{1}{1 − α}u_{2}) = sup (\frac{1}{α}u_{1}, \frac{1}{1 − α}u_{2}) (0 ≤ α ≤ 1).

Formula (6) shows that u_{1} + u_{2} = \overline{inf}_{0≤α≤1}w_{α}. Since u_{1} + u_{2} ∈ U it follows that \overline{inf}_{0≤α≤1}w_{α} ∈ U. Therefore

u_{1} ˆ+u_{2} = inf_{0≤α≤1} w_{α} = \overline{inf}_{0≤α≤1}w_{α} = u_{1} + u_{2}.

2) Applying (7) we see that

u_{1} ⊕ u_{2} = sup_{0≤α≤1} inf(αu_{1}, (1 − α)u_{2}) = \overline{sup}_{0≤α≤1}\overline{inf}(αu_{1}, (1 − α)u_{2}) = \frac{1}{\frac{1}{u_{1}} + \frac{1}{u_{2}}} .

Corollary 4.4. Let U and V be c^{2}-lattices and P : U → V be an antihomogeneous conjugacy operator. Then

1. If V ∈ U and V contains \overline{inf}(v_{1}, v_{2}) whenever v_{1}, v_{2} ∈ V (for example V is one of the following c^{2}-lattices : F (X), M_{0}(X), M^{0}(X), P H(X) then the operator P maps a “sum” u_{1} ˆ+u_{2} to the inverse sum of the functions P (u_{1}) and P (u_{2}).

2. If V ∈ U and V contains the pointwise sum v_{1} +v_{2} whenever v_{1}, v_{2} ∈ V (for example V is any one of the following c^{2}-lattices : F (X), M_{0}(X), M^{0}(X), P H(X), SL(X), CF (X) or c^{2}-lattice C_{1}^{H}(Y) of H-convex functions) then the operator P maps the “inverse sum” u_{1} ⊕ u_{2} to the pointwise sum of the functions P (u_{1}) and P (u_{2}).

3. If U ∈ U and U contains the pointwise sum u_{1} + u_{2} whenever u_{1}, u_{2} ∈ U then the operator P maps a pointwise sum u_{1} + u_{2} to the “inverse sum” P (u_{1}) ⊕ P (u_{2}) .

Let us give some examples. At first we consider c^{2}-lattices of sets. Let A, B be subsets of a locally convex Hausdorff space X. We will use the following notation:

A + B = {a + b : a ∈ A, b ∈ B}, A ˆ⊕B = ⋃_{0≤α≤1}[(αA) ∩ ((1 − α)B)].

It is easily seen that if A, B are closed star with regards to zero sets then A ˆ⊕B also is closed and star with regards to zero, convexity of A, B implies convexity of A ˆ⊕B . Let us note that operation ˆ⊕ is well known in Convex Analysis (see for example [1]). This operation was studied in [18] for special class of star with regards to zero sets. Below we consider dual vector spaces X and L.

Example 4.5. Let ST (X) be a c^{2}-lattice of all closed star with regards to zero subsets of the space X. Clearly if A, B ∈ ST (X) then

A ˆ+B = ⋂_{0≤α≤1}[(\frac{1}{α}A) ∪ (\frac{1}{1 − α}B)], A ⊕ B = A ˆ⊕B.

Example 4.6. Let CS(X) be a c^{2}-lattice of all closed convex containing zero subsets of the space X. We have by definition

A ˆ+B = ⋂_{0≤α≤1} conv[(\frac{1}{α}A) ∪ (\frac{1}{1 − α}B)], A ⊕ B = A ˆ⊕B.

where convA is a convex hull of the set A. On the other hand we can show that the next assertion is true.

Proposition 4.7. If A, B ∈ CS(X) then A ˆ+B = cl(A + B) .

Proof. We apply the isomorphism φ of c^{2}-lattices CS(X) and SL(L) where φ(A) is the support function p_{A} of the set A. By definition p_{A}(l) = sup_{x∈A}[l, x]. Since φ is isomorphism it follows that

φ(inf_{α} A_{α}) = inf_{α} φ(A_{α}),  φ(sup_{α} A_{α}) = sup_{α} φ(A_{α}),  φ(λA) = λφ(A) ∀λ > 0.

We have

p_{A ˆ+B} = φ(A ˆ+B)

= φ(inf_{α} sup (\frac{1}{α}A, \frac{1}{1 − α}B))

= inf_{α} sup (\frac{1}{α}φ(A), \frac{1}{1 − α}φ(B))

= inf_{α} sup (\frac{1}{α}p_{A}, \frac{1}{1 − α}p_{B})

= p_{A} ˆ+p_{B} = p_{A} + p_{B}.

It is well known that p_{A} + p_{B} = p_{cl(A+B)} . Therefore A ˆ+B = cl(A + B) .

Corollary 4.8. If A, B ∈ CS(X) and if P : CS(X) → CS(L) is an antihomogeneous conjugacy operator then P (cl(A+B)) = P (A) ˆ⊕P (B) and P (A ˆ⊕B) = cl(P (A)+P (B)) .

Example 4.9. Let us consider the c^{2}-lattice SL(X). Proposition 4.3 shows that p + q = p ˆ+q for all p, q ∈ SL(X).

Proposition 4.10. If A, B ∈ SL(X) then

p ⊕ q = sup_{0≤α≤1}(αp) ˜⊕((1 − α)q)

where f ˜⊕g is the inf-convolution of the functions f and g.

Proof. Let ∂p = {x ∈ X : [l, x] ≤ p(l) ∀l ∈ L}, the subdifferential of a sublinear function p. Clearly ∂p = φ^{−1}(p) where the mapping φ was defined in the proof of the proposition 4.7. We have

∂(p ⊕ q) = φ^{−1}(p ⊕ q) = φ^{−1}(sup_{α} inf(αp, (1 − α)q)) =

sup_{α} inf(αφ^{−1}(p), (1 − α)φ^{−1}(q)) = sup_{α} inf(α∂p, (1 − α)∂q) = ∂p ⊕ ∂q = ∂p ˆ⊕∂q.

It is well known ( see for instance [1]) that ∂p ˆ⊕∂q = ∂s where s = sup_{0≤α≤1}(αp) ˜⊕(1 − α)q. Therefore p ⊕ q = s.

Example 4.11. Let us compute a set whose gauge coincides with the sum of gauges of given sets.

Proposition 4.12. Let A, B ∈ ST (X) and μ_{A} be the gauge of A, and μ_{B} be the gauge of B. Then μ_{A} + μ_{B} is a gauge of the set A ˆ⊕B.

Proof. Let us consider the antihomogeneous conjugacy operator P : P H(X) → ST (X) where P (μ) = {x : μ(x) ≤ 1}. Clearly P (μ_{A}) = A, P (μ_{B}) = B. We have

P (μ_{A} + μ_{B}) = P (μ_{A}) ⊕ P (μ_{B}) = A ⊕ B = A ˆ⊕B

therefore the gauge of A ˆ⊕B is equal to the sum μ_{A} + μ_{B}.

Example 4.13. Now let us consider the c^{2}-lattice C\overline{H} ; 1(Y) of \overline{H}-convex functions

defined on the cone Y and the c^{2}-lattice C\overline{Y} ; 2 (H) of \overline{Y} -convex functions defined on the cone H (see Example 3.8 of Section 3) and the antihomogeneous conjugacy operator

P : C\overline{H} ; 1(Y) → C\overline{Y} ; 2 (H) defined by formula (3). We have

P (f_{1} + f_{2})(x) = (P (f_{1}) ⊕ P (f_{2}))(x) = sup_{0≤α≤1} inf(α ⊗ P (f_{1}), (1 − α) ⊗ P (f_{2}))(x)

= sup_{0≤α≤1} co_{H} \overline{inf} [\frac{1}{α}P (f_{1})(αx), \frac{1}{1 − α}P (f_{2})((1 − α)x)].

In particular if f_{1}, f_{2} are l.s.c convex functions and P (f) coincides with the conjugate function (in the sense of Convex Analysis) f^{∗} then

(f_{1} + f_{2})^{∗} = sup_{0≤α≤1} conv \overline{inf} [\frac{1}{α}f_{1}^{∗}(αx), \frac{1}{1 − α}f_{2}^{∗}(1 − α)x)]

where convg is the convex regularization of the function g i.e. convg(x) = sup{a(x) : a ≤ g, a is an affine function}. By the famous Moreau -Rockafellar theorem the equality (f_{1} + f_{2})^{∗} = conv(f_{1}^{∗} ˜⊕f_{2}^{∗}) holds. Therefore the following is true.

Proposition 4.14. If f_{1}, f_{2} are convex l.s.c. functions then

conv(f_{1} ˜⊕f_{2})(x) = sup_{0≤α≤1} conv \overline{inf}[\frac{1}{α}f_{1}(αx), \frac{1}{1 − α}f_{2}((1 − α)x)].

Let us discuss Proposition 4.14. Clearly the inf-convolution is a global operation. In other words we must know quantities f_{1}(y) and f_{2}(y) for all y ∈ X in order to compute the function f_{1} ˜⊕f_{2} at the given point x ∈ X. Proposition 4.14 shows that the main reason for the global nature of inf-convolution is the use of the convex hull of a nonconvex function. If conv(f_{1} ˜⊕f_{2})(x) = (f_{1} ˜⊕f_{2})(x) and if we can apply the convex hull for computing the function f_{1} ˜⊕f_{2} at the point x then we have to know only the quantities f_{1}(αx) and f_{2}(αx) for 0 ≤ α ≤ 1.

5. Properties of the operations ˆ+ and ⊕

Let U be a c^{2}-lattice with the following properties:

i) If 0 < λ < +∞ and Ω is a subset of U then

sup{λu : u ∈ Ω} = λ sup{u : u ∈ Ω}, inf{λu : u ∈ Ω} = λ inf{u : u ∈ Ω}.

ii) If 0 < λ, μ < +∞ and u ∈ U then

sup(λu, μu) = sup(λ, μ)u, inf(λu, μu) = inf(λ, μ)u.

Let us study some algebraic properties of the operations ˆ+ and ⊕ defined by formulae (4) and (5) accordingly. Clearly these operations are commutative:

u ˆ+v = v ˆ+u, u ⊕ v = v ⊕ u.

Now let us clear up the connections between multiplication by a positive scalar and the operations ˆ+ and ⊕. We have with 0 < λ < +∞

λu ˆ+λv = inf_{0≤α≤1} sup (\frac{λ}{α}u, \frac{λ}{1 − α}v) = λ(u ˆ+v),

λu ⊕ λv = sup_{0≤α≤1} inf(λαu, λ(1 − α)v) = λ(u ⊕ v).

Applying (6) we have also

λu ˆ+μu = inf_{0≤α≤1} sup (\frac{λ}{α}u, \frac{μ}{1 − α}u) = inf_{0≤α≤1} sup (\frac{λ}{α}, \frac{μ}{1 − α}) u = (λ + μ)u. (8)

In the same manner we can see applying (7) that

λu ⊕ μu = \frac{1}{\frac{1}{λ} + \frac{1}{μ}} u.  (9)

We see that the c^{2}-lattice U generates two algebraical systems. The first of them is U with operations ˆ+ and · , the second is U with operations ⊕ and ⊙ where λ ⊙ u = \frac{1}{λ} · u. We showed that commutative and distributive laws hold in these systems. Clearly a conjugacy operator P : U → V is an isomorphism between (U, ˆ+, ·) and (V, ⊕, ⊙). Unfortunately associativity does not hold without additional assumptions in the algebraical systems under consideration. Let us give an example.

Example 5.1. Let us consider the c^{2}-lattice Q(IR^{2}) of all nonnegative l.s.c. vanishing at zero and quasiconvex functions defined on the plane IR^{2} (see Situation 1 of Section 2). We set for x = (x_{1}, x_{2})

q_{1}(x) = \{ \begin{matrix}1 & if x_{1} ≥ 1 \\ 0 & if x_{1} < 1 \end{matrix}

q_{2}(x) = \{ \begin{matrix}1 & if x_{2} ≥ 1 \\ 0 & if x_{2} < 1 \end{matrix}

q_{3}(x) = \{ \begin{matrix}+∞ & x ≠ λ(1, 1) (λ ≥ 0) \\ λ & x = λ(1, 1) (λ ≥ 0) \end{matrix}

Clearly q_{i} ∈ Q(IR^{2}), i = 1, 2, 3. Let E = {x = (x_{1}, x_{2}) : x_{1} ≤ 1, x_{2} ≤ 1}. Since (q_{1} ˆ+q_{2}) = co_{Q(IR^{2})}(q_{1} + q_{2}) we can compute that

(q_{1} ˆ+q_{2})(x) = \{ \begin{matrix}1 & x ∈ R^{2} \ E \\ 0 & x ∈ E \end{matrix}

Therefore q_{1} ˆ+q_{2} ≠ q_{1} +q_{2}. On the other hand (q_{1} ˆ+q_{2}) ˆ+q_{3} = (q_{1} ˆ+q_{2})+q_{3} and q_{1} ˆ+(q_{2} ˆ+q_{3}) = q_{1} + (q_{2} + q_{3}). Applying these equalities we can show that

(q_{1} ˆ+q_{2}) ˆ+q_{3} ≠ q_{1} ˆ+(q_{2} ˆ+q_{3}).

There are many c^{2}-lattices where the operation ˆ+ and ⊕ are associative. By proposition 3 we have: if U ∈ U and u_{1} + u_{2} ∈ U whenever u_{1}, u_{2} ∈ U then associativity holds for operation ˆ+; if U ∈ U and \overline{inf}(u_{1}, u_{2}) ∈ U whenever u_{1}, u_{2} ∈ U then associativity holds for operation ⊕. Let us give one more condition which guarantees associativity.

Proposition 5.2. If U is a c^{2}-lattice with distributive laws: for all u ∈ U and for an arbitrary family (v_{ξ})_{ξ} ∈ U the following equalities hold

inf(u, sup_{ξ} v_{ξ}) = sup_{ξ} inf(u, v_{ξ})  (10)

sup(u, inf v_{ξ}) = inf_{ξ} sup(u, v_{ξ})  (11)

then operations ˆ+ and ⊕ are associative.

Proof. We first compute (u ⊕ v) ⊕ w. Let us take the formula (10) for this purpose. We have

(u ⊕ v) ⊕ w = sup_{0≤γ≤1} inf(γ(u ⊕ v), (1 − γ)w) =

sup_{0≤γ≤1} inf[ sup_{0≤α≤1} inf(γαu, γ(1 − α)v), (1 − γ)w] =

sup_{0≤γ≤1} sup_{0≤α≤1} inf(γαu, γ(1 − α)v, (1 − γ)w) =

sup_{λ,μ,ν≥0;λ+μ+ν=1} inf(λu, μv, νw).

In the same manner we can see that

u ⊕ (v ⊕ w) = sup_{λ,μ,ν≥0;λ+μ+ν=1} inf(λu, μv, νw).

Therefore associativity holds for the operation ⊕. Applying (11) we obtain associativity for the operation ˆ+.

Proposition 5.3. If U, V are c^{2}-lattices and if there is an antihomogeneous conjugacy operator P : U → V then the following are equivalent: i) associativity holds for operation ˆ+ in the U, ii) associativity holds for operation ⊕ in the V.

Proof. This is clear.

Let us give some examples.

Example 5.4. Operation ⊕ is associative in the c^{2}-lattices SL(X), CF (X), CS(X), and ST (X).

Example 5.5. Since there is an antihomogeneous conjugacy operator P : Q(X) → Q(X) operation ⊕ is not associative in the Q(X).

Example 5.6. Let U be a c^{2}-lattice such that at least one of the operations ˆ+ and ⊕ is associative. Then there is no antihomogeneous conjugacy operator P : Q(X) → U.

6. Operator of polarity

In this section we consider the operator of polarity π : CS(X) → CS(X′) where X is a locally convex Hausdorff space with conjugate space X′. We consider X and X′ as dual vector spaces, in particular we assume that weak^{∗} topology is introduced in the space X′. By definition π(A) = A^{∘} where A^{∘} is the polar set of a set A. Let T : X → X be a weakly continuous linear operator. It is well known that

π ∘ T = (T^{∗})^{−1} ∘ π

where T^{∗} is the conjugate operator with regard to T . So we are interested in operators P : CS(X) → CS(X′) such that

P ∘ T = (T^{∗})^{−1} ∘ P.  (12)

As it turns out an antihomogeneous conjugacy operator P : CS(X) → CS(X′) is the operator of polarity if equality (12) holds only for the class of simplest operators T and besides there is the unique set A such that P (A) = π(A). Let l ∈ X′ and u ∈ X such that [l, u] = 1. Set T_{l,u}(x) = x − [l, x]u for all x ∈ X. Clearly T_{l,u} is a projection from X on hyperplane H = l^{−1}(0). On the other hand if H = l^{−1}(0) is a hyperplane and T is a projection from X on H then there is a vector u ∈ X such that [l, u] = 1 and T = T_{l,u}. Let τ be the set of all projections from X on closed hyperplanes. We have the following:

Proposition 6.1. Let P : CS(X) → CS(X′) be an antihomogeneous conjugacy operator. Assume that (12) is true for all T ∈ τ ∪ {−Id} where Id is the identity and there is a nonzero vector v ∈ X such that P (s[0, v]) = π(s[0, v]) where s[0, v] is the segment {λv : 0 ≤ λ ≤ 1}. Then P = π.

Proof. Let us consider w ∈ X, w ≠ λv for all real λ and an operator T ∈ τ such that T (v) = w. For example T (x) = x − [l, x](v − w) with L ∈ X′, [l, v] = 1, [l, w] = 0. We have

P (s[0, w]) = P (T (s[0, v]) = (T^{∗})^{−1}P (s[0, v]) = (T^{∗})^{−1}π(s[0, v]) = π(s[0, w]).

In the same manner we can see that P (s[0, −v]) = π(s[0, −v]). Since P and π are antihomogeneous conjugacy operators we have P (s[0, λv]) = π(s[0, λv]) for all real λ. Now let A ∈ CS(X). We have A = ∪_{x∈A}[0, x] and therefore

P (A) = ⋂_{x∈A=} P (s[0, x]) = ⋂_{x∈A} π(s[0, x]) = π(A).

Sometimes we can consider a more interesting set instead of a segment s[0, v]. Let H be a Hilbert space. It is well known that the unit ball B is the unique fixed point of the polarity operator π : CS(H) → CS(H). An antihomogeneous conjugacy operator P : CS(H) → CS(H) with the property P (B) = π(B) has been considered in the paper [19] and conditions which guarantee either equality P = π or equality P = −π were given in that paper. Unfortunately there is a minor defect in the proof of the theorem in [19]. Below we will consider a set B in a locally convex Hausdorff space X with the following properties:

i) B is a symmetrical closed convex set and intB ≠ ∅; ii) there is a linear function l ∈ X′ and projection T_{0} from X on H = l^{−1}(0) such that

T_{0}(B) ⊂ B; iii) there is an element y ∈ B such that [l, y] = sup_{x∈B}[l, x]. Clearly the unit ball of a Hilbert space possesses properties i) - iii); there are many sets with properties i) - iii) in a finite - dimensional space. A strip {x : −1 ≤ [l, x] ≤ 1} in the locally convex space X where l ∈ X′ has these properties also. We will denote by τ_{1} the set of all linear projections from X on a straight line. Clearly T ∈ τ_{1} if and only if there are l ∈ X′ and u ∈ X such that T (x) = [l, x]u.

Theorem 6.2. Let P : CS(X) → CS(X′) be an antihomogeneous conjugacy operator. Assume that the formula (12) is true for all T ∈ τ ∪ τ_{1} ∪ {−Id} and there is a subset B of the space X such that P (B) = π(B) and properties i) - iii) above hold. Then either P = π or P = −π, where −π(A) = −A^{∘}

Proof. Let the projector T_{0} (see ii) above) have a form T_{0}(x) = x−[l, x]u with [l, u] = 1. We will prove that either P (s[0, u]) = π(s[0, u]) or P (s[0, u]) = −π(s[0, u]) and then apply Proposition 6.1. The proof will be divided into four steps.

1. Let us consider the sets C = {x ∈ B : [l, x] = 0} = B∩H and C′ = {f ∈ B^{∘} : [f, u] = 0} = B^{∘} ∩ H′ . Here H′ = u^{−1}(0). We first show that the equality P (C) = C′ + IRl holds where IRl is a line (λl)_{−∞<λ<+∞}. It is easy to check that the conjugate with regard to T_{0}, operatorT_{0}^{∗}, is a projection from X′ on H′ which has the form T_{0}^{∗}(f) = f − [f, u]l . It is easily seen that (T_{0}^{∗})^{−1}(f) = f + IRl. Let us check the equality T_{0}(B) = C. Applying inclusions T_{0}(B) ⊂ B and T_{0}(B) ⊂ H we obtain T_{0}(B) ⊂ C. On the other hand inclusion B ⊃ C implies T_{0}(B) ⊃ T_{0}(C) = C. Since T_{0}(B) = C and T_{0} ∈ τ we have

P (C) = (T_{0}^{∗})^{−1}P (B) = (T_{0}^{∗})^{−1}(B^{∘}) = (T_{0}^{∗})^{−1}(C′) = C′ + IRl.

2. Let W = {x ∈ B : [l, x] ≥ 0} and

Z_{+} = B^{∘} + IR_{+}l,  Z_{−} = B^{∘} − IR_{+}l.  (13)

where IR_{+}l is the ray (λl)_{λ≥0}. Since B is a symmetrical set it follows that B^{∘} is a symmetrical set too. Therefore Z_{−} = −Z_{+}. Now we prove that either P (W) = Z_{+} or P (W) = Z_{−}. We will denote P (W) by Z. Since B is a symmetrical set we have −W = {x ∈ B : [l, x] ≤ 0}. Therefore

W ∪ (−W) = B W ∩ (−W) = C.

On the other hand P (−W) = P ((−Id)W) = −Z. As P is an antihomogeneous conjugacy operator and P (C) = C′ + IRl we have the next system of two equations with regard to an unknown convex w^{∗}-closed set Z:

Z ∩ (−Z) = B^{∘}  (14)
cl conv(Z ∪ (−Z)) = C′ + IRl.  (15)

We will show that there are only two solutions of the system (14) - (15): one of them is Z_{+} and the other is Z_{−} = −Z_{+}, where Z_{+}, Z_{−} are defined by formula (13). First of all let us note that assumption intB ≠ ∅ implies w^{∗} -compactness of the set B^{∘} and therefore the set C′ is w^{∗} -compact. Let K be the recession cone of the set Z. Applying (15) and weak* - compactness of C′ we conclude that K ⊂ IRl. Clearly −K is a recession cone of (−Z) so (14) shows that K ≠ IRl. Our next goal is to show that K ≠ {0}. Let q denote the function defined on the set C′ by formula

q(f) = sup{λ : f + λl ∈ Z}

Since Z is a convex w^{∗} -closed set it follows that q is a concave and w^{∗} -upper semicontinuous function. Assume that K = {0}. We have q(f) < +∞ for all f ∈ C′ in this case. The w^{∗} -compactness of the set C′ implies that the functuon q is bounded from above on this set. Thus

sup_{f∈C′} sup{λ : f + λl ∈ Z} < +∞.

In the same manner we can conclude that

sup_{f∈C′} sup{λ : f + λl ∈ (−Z)} < +∞.

From (15) we obtain that cl conv(Z ∪ (−Z)) ⊂ C′ + IRl. Since [f, u] = 0 ∀f ∈ C′ and [l, u] = 1 we have

sup{[g, u] : g ∈ cl conv(Z ∪ (−Z)}
= sup{[g, u] : g ∈ (Z ∪ (−Z)}
= sup{[f + λl, u] : f ∈ C′, f + λl ∈ Z ∪ (−Z)}
= sup{λ : f ∈ C′, f + λl ∈ Z ∪ (−Z)} < +∞  (16)

On the other hand applying (15) we can assert that

sup{[g, u] : g ∈ cl conv(Z ∪ (−Z))} = sup{[g, u] : g ∈ C′ + IRl} = +∞

which contradict the formula (16).

Thus K ⊂ IRl, K ≠ IRl and K ≠ {0}. Therefore K is either IR_{+}l or −IR_{+}l. Assume that K = IR_{+}l. It follows from (14) that B^{∘} ⊂ Z. By the definition of recession cone we have

Z_{+} = B^{∘} + IRl ⊂ Z + IRl = Z

Let us show that Z_{+} = Z in this case. Let f ∈ Z. Applying (15) we can find g ∈ C′ and a real μ such that f = g + μl. If μ ≥ 0 then f ∈ C′ + IR_{+}l ⊂ B^{∘} + IR_{+}l = Z_{+}. Now let us assume μ < 0. We have in this case

−f = −g + (−μ)l ⊂ C′ + IR_{+}l ⊂ Z_{+} ⊂ Z.

Therefore f ∈ −Z and f ∈ (−Z) ∩ Z. Applying (14) we have

f ∈ B^{∘} ⊂ B^{∘} + IR_{+}l = Z_{+}.

The equality Z = Z_{+} is proved. In the same manner we can see that Z = Z_{−} if K = −IR_{+} 3. Let us compute π(W). We will denote by H_{+} the half-space {x : [l, x] ≥ 0} of the space X. Clearly π(H_{+}) = −IR_{+}l . Since W = B ∩ H_{+} we have

π(W) = conv(B^{∘} ∪ (−IR_{+}l)) = B^{∘} − IR_{+}l = Z_{−}.

It follows that either P (W) = π(W) or P (W) = −π(W).

4. Let T_{∗} be the projection operator from X on the line Ru which is defined by formula T_{∗}(x) = [l, x]u. If P (W) = π(W) then

P (s[0, λu]) = P (T_{∗}(W)) = T_{∗}^{−1}P (W) = T_{∗}^{−1}π(W) = π(T_{∗}(W)) = π(s[0, λu]).

In the same manner we can see that P (s[0, λu]) = −π(s[0, λu]) if P (W) = −π(W). Now we can use either Proposition 6.1 itself or a version of this proposition for the operator −π in order to complete the proof.

Acknowledgment. The author wishes to thank Dr. B. M. Glover and two anonymous referees for valuable comments which have improved this paper.

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