Journal of Convex Analysis Volume 7 (2000), No. 1, 95–113

Harmonic Sum and Duality

Jean-Paul Penot

Université de Pau, Faculté des Sciences, Département de Mathématiques, 64000 Pau, France. e-mail: jean-paul.penot@univ-pau.fr

Constantin Zălinescu^{∗}

Université “Al. I. Cuza” Iaşi, Faculté de Mathématiques, Bd. Copou Nr. 11, 6600 Iaşi, Romania. e-mail: zalinesc@uaic.ro

Received April 24, 1999 Revised manuscript received December 23, 1999

We consider an operation on subsets of a topological vector space which is closely related to what has been called the inverse addition by R.T. Rockafellar. Applied to closed convex sets, it appears as the operation corresponding to the addition under polarity. However, our study is not limited to the convex case. Crucial tools for it are the gauges one can associate with a subset. We stress the role played by asymptotic cones in such a context. We present an application to the calculus of conjugate functions for one of the most fruitful dualities for quasiconvex problems. We also present an extension of the well-known rule for the computation of the normal cone to a convex set defined by a convex inequality.

Keywords: conjugate function, convex sets, duality, gauge, harmonic sum, inverse sum, normal cone, shady set, star-shaped set

1991 Mathematics Subject Classification: 52A05, 52A30, 26B25, 90C25

1. Introduction

The use of gauges for the study of convex sets is well established after the work of Minkowski during the first decade of this century (see classical monographs on functional analysis and convex analysis) and the recent works of Barbara-Crouzeix ([4]), Rubinov and his co-authors ([21]-[24]). It is the purpose of the present paper to show the usefulness of such a tool for the study of an operation on the family of subsets of a topological vector space which is akin to the operation called the inverse sum (see [18], [19]). Here we modify the formula defining the inverse sum in order to get a closed subset when the given sets are closed. To reach this aim one needs to take into account the asymptotic cones or some substitutes. This aim has been achieved by Ioffe-Tikhomirov [6], Rockafellar [18] and Rubinov-Yagubov [24] when the sets are closed, convex and contain 0 (or are star-shaped). Here we consider more general cases by using explicitly asymptotic cones. Moreover, we also modify the classical definitions of the gauges associated with a subset in order to obtain semicontinuity properties. In the particular case of closed shady (or star-shaped at infinity [8], [9], [22]) convex subsets not containing 0 (and even for a larger class of subsets) our proposals coincide with the functions obtained via regularization in [4]. The simultaneous uses of these modifications enable one to get pleasant relationships

∗The contribution of this author has been accomplished during his stay at Université de Pau, spring 1999.

ISSN 0944-6532 / $ 2.50 \begin{matrix}c \\ ◯ \end{matrix} Heldermann Verlag

96 J.-P. Penot, C. Zălinescu / Harmonic sum and duality

between the set-theoretical operation of inverse sum (or rather its variant considered here, we call the harmonic sum) and sums of gauges. These relationships are especially useful when considering duality questions.

The terminology we use is justified by the following simple case of two closed intervals A := [ \underline{a}, \overline{a}], B := [ \underline{b}, \overline{b}] with 0 ≤ \underline{a} ≤ \overline{a} < ∞ and 0 ≤ \underline{b} ≤ \overline{b} < ∞; then the inverse sum A#B and the harmonic sum A◇B are given by

A#B = A◇B = [(\underline{a}^{−1} + \underline{b}^{−1})^{−1}, (\overline{a}^{−1} + \overline{b}^{−1})^{−1}] .

For A := [ \underline{a}, ∞[, B := [ \underline{b}, \overline{b}] one has

A#B = [(\underline{a}^{−1} + \underline{b}^{−1})^{−1}, \overline{b}[ , A◇B = [(\underline{a}^{−1} + \underline{b}^{−1})^{−1}, \overline{b} ] .

This simple example shows one has to modify the definition of the inverse sum in order to get a closed set even when A and B are closed convex subsets.

As an application, we show that the harmonic sum enters in the computation of the conjugate of a quasiconvex function for one of the most important duality schemes of quasiconvex analysis (see [5], [10], [12], [14], [15], [31]...). In general, explicit formulae for such conjugates are not easy to obtain. Here our computation strongly relies on a convex duality result. A by-product of this duality result is a refined form of the representation theorem for the normal cone to a convex set defined by a convex inequality under a Slater condition.

Let us note that the formulae we obtain for this quasiconvex conjugate show that it is useful to consider gauges of sets which are neither shady nor star-shaped. Another such situation appears in the calculus of the classical subdifferentials specific to quasiconvex functions. This question (which motivated the present study) is considered in the paper [16].

Acknowledgements. We are indebted to an anonymous referee for the suggestion of putting apart the contents of the present paper from [16] and for several references related to the subject. We are also indebted to another referee for pointing out the links of the present paper with a previous study by A. Seeger ([25]). There the harmonic sum is included in a general spectrum of operations on convex sets containing the origin; that paper also contains such operations on real-valued functions and applications are given to second order subdifferentials of convex functions and to network connections.

2. Gauges

Henceforth we consider X to be a topological vector space. In the sequel, we say that a subset A of X is star-shaped with respect to 0, or in short is star-shaped, if 0 ∈ A and ]0, 1] · A := {ta | 0 < t ≤ 1, a ∈ A} ⊂ A. We say that A is shady if [1, ∞[A ⊂ A (see [8], [9], [22], [23]). For a nonempty subset A of X we use the convention that 0·A = 0A = 0^{+}A := lim sup_{t→0_{+}} tA, the asymptotic cone of A, i.e. the set of limits of nets of the form (t_{i}a_{i})_{i∈I} with (t_{i})_{i∈I} → 0, t_{i} ∈ ℙ := ]0, ∞[, a_{i} ∈ A for each i ∈ I. We observe that this convention ensures that [0, γ] · A is closed for every γ ∈ ℝ_{+} := [0, ∞[ when A is closed. If A = ∅ we consider that 0^{+}A = {0}, as in [20]. In the sequel, for the empty set in ℝ, we use the familiar conventions inf ∅ = +∞ and sup ∅ = −∞.

Note that, when 0 ∈ A, Ioffe-Tikhomirov [6] and Rubinov [21, p. 296] took 0 · A as being ∩_{λ>0}λA; if A is a closed and star-shaped set, one obtains again the asymptotic cone of A.

Recall that the Minkowski gauge functional of a subset A of X is given by

μ_{A} : X → \overline{ℝ}, μ_{A}(x) := \begin{cases}inf\{t > 0 | x ∈ tA\} & if x ≠ 0, \\ 0 & if x = 0, \end{cases}

where \overline{ℝ} := ℝ ∪ \{−∞, +∞\}. Here we do not suppose 0 ∈ A as it is done usually; we will see in Proposition 5.2 that there are natural situations in which 0 ∉ A.

Another functional associated to A, considered and studied by Barbara-Crouzeix in [4] under the name of concave gauge function, is given by

ν_{A} : X → \overline{ℝ}, ν_{A}(x) := \begin{cases}sup\{t > 0 | x ∈ tA\} & if x ≠ 0, \\ 0 & if x = 0. \end{cases}

Note that μ_{B} ≤ μ_{A} and ν_{A} ≤ ν_{B} for A ⊂ B, μ_{A∪B} = μ_{A} ∧ μ_{B} := min (μ_{A}, μ_{B}), ν_{A∪B} = ν_{A} ∨ ν_{B} := max (ν_{A}, ν_{B}). Also note that μ_{A} = μ_{]0,1]·A} = μ_{A∪{0}} and ν_{A} = ν_{A\{0}} = ν_{[1,∞[·A}.

Example 2.1. Note that it may happen that μ_{A} and −ν_{A} are not lower semicontinuous, even if A is closed and convex. For example, if A = {(r, s) ∈ ℝ^{2} | r, s > 0, rs ≥ 1}, then

μ_{A}(r, s) = \begin{cases}0 & if (r, s) ∈ ℙ^{2} ∪ \{(0, 0)\}, \\ ∞ & otherwise, \end{cases} ν_{A}(r, s) = \begin{cases}\sqrt{rs} & if (r, s) ∈ ℙ^{2} ∪ \{(0, 0)\}, \\ −∞ & otherwise. \end{cases}

The preceding example justifies the introduction of the following functionals (in the definition of which we use the above convention about 0A):

α_{A} : X → \overline{ℝ},  α_{A}(x) := inf\{t ≥ 0 | x ∈ tA\},
β_{A} : X → \overline{ℝ},  β_{A}(x) := sup\{t ≥ 0 | x ∈ tA\}.

Thus, the domains of μ_{A} and −ν_{A} are ℙA∪{0}, where ℙA =: cone A is the cone generated by A. Let us observe that the functions μ_{A} and α_{A} differ only on 0^{+}A \ (0_{r}^{+}A ∪ {0}), where 0_{r}^{+}A := ⋂_{γ>0}]0, γ]A is the radial asymptotic cone of A and the functions ν_{A} and β_{A} are different only on 0^{+}A \ ℙA. Let us also note that in the definition of ν_{A}(x) (resp. β_{A}(x)) the supremum is attained when it is finite and A is centrally closed in the sense that for any x ∈ X, (t_{n}) → 1, one has x ∈ A whenever t_{n}x ∈ A for each n. A similar observation can be made for μ_{A}(x) and α_{A}(x).

Of course, if A ⊂ B then α_{B} ≤ α_{A}, β_{A} ≤ β_{B}. Note that β_{A}, as well as ν_{A}, does not take the value +∞ if 0 ∉ clA or if 0 ∉ A and A is centrally closed.

The following relations, in which γ ∈ ℙ, are obvious:

{x ∈ X | μ_{A}(x) < γ} ⊂ (]0, γ] · A) ∪ {0} ⊂ {x ∈ X | μ_{A}(x) ≤ γ},  (2.1)
{x ∈ X | ν_{A}(x) > γ} ⊂ [γ, ∞[·A ⊂ {x ∈ X | ν_{A}(x) ≥ γ}.  (2.2)

The four functions are positively homogeneous and if A is convex, μ_{A} is sublinear and, when the cone ℙA is pointed, −ν_{A} is sublinear, too. When A is a closed convex subset, then α_{A} is convex, hence sublinear, and if moreover ℝ_{+}A is pointed, β_{A} is concave. Here we use the familiar convention from convex analysis (+∞)+(−∞) = +∞. The preceding discussion can be subsumed in the following statement which completes results in [22], [24].

Proposition 2.2. The set of centrally closed convex subsets of X not containing 0 is in bijection with the set of pairs (μ, ν), where μ and −ν are extended real-valued sublinear functions on X, such that dom μ = dom(−ν) and 0 ≤ μ ≤ ν on dom μ. The inverse of the mapping A ↦ (μ_{A}, ν_{A}) is the mapping which assigns to (μ, ν) the set A := {x ∈ X : μ(x) ≤ 1 ≤ ν(x)} .

When one restricts one’s attention to the family St(X) of centrally closed star-shaped subsets of X or to the family Sh(X) of centrally closed convex shady subsets of X not containing 0, it suffices to use A ↦ μ_{A} or A ↦ ν_{A}, respectively (see [22], [24]).

For star-shaped subsets and shady subsets nice characterizations of the asymptotic cone in terms of the gauges introduced above are available. For A closed convex containing 0, the relation 0^{+}A = μ_{A}^{−1}(0), obtained from part (a) (taking assertions (c) and (d) into account) of the following proposition, is well known (see [20] p. 101 for example).

A relationship between μ_{A} and α_{A} is given in the following proposition. For an extended real-valued function f on X we denote by \overline{f} the lower semicontinuous (lsc) hull of f and by \underline{f} the upper semicontinuous (usc) hull of f.

Proposition 2.3. Let A be a subset of X. Then:

(a) α_{A} = α_{]0,1]A} = α_{[0,1]A} and 0^{+}A = {x ∈ X | α_{A}(x) = 0};

(b) for each γ ∈ [0, ∞[ one has

[0, γ] · A ⊂ {x ∈ X | α_{A}(x) ≤ γ} ⊂ [0, γ] · cl A.  (2.3)

Thus, if A is closed one has [0, γ] · A = {x ∈ X | α_{A}(x) ≤ γ} and α_{A} is lsc;

(c) α_{cl A} ≤ α_{A} ≤ μ_{A} and \overline{μ_{A}} = \overline{α_{A}} = α_{cl A}.

(d) If A = ]0, 1]A ≠ ∅ one has \overline{μ_{A}} = μ_{cl A}. Moreover,

cl A = cl{x ∈ X | μ_{A}(x) < 1} = cl{x ∈ X | μ_{A}(x) ≤ 1}.  (2.4)

In particular, if A is closed and star-shaped, then μ_{A} = α_{A} and A = μ_{A}^{−1}([0, 1]).

Proof. (a) For the first equalities it is sufficient to observe that 0^{+}A = 0^{+} (]0, 1]A) = 0^{+} ([0, 1]A) . The inclusion 0^{+}A ⊂ α_{A}^{−1}(0) is obvious. Conversely, given x ∈ α_{A}^{−1}(0), for each n ∈ ℕ^{∗} there exist t_{n} ∈ [0, 1/n] and a_{n} ∈ A such that x = t_{n}a_{n}. If t_{n} = 0 for some n then x ∈ 0^{+}A. If t_{n} > 0 for every n ∈ ℕ, then x ∈ 0^{+}A.

(b) The first inclusion in relation (2.3) is obvious. Let x be such that t := α_{A}(x) ≤ γ. Then there exists a sequence (t_{n}) ∈ [0, ∞[ converging to t such that x ∈ t_{n}A for each n ∈ ℕ. If t = 0, as above, we have that x ∈ 0^{+}A ⊂ [0, γ] · A. If t > 0 we have (t_{n}^{−1}x) → t^{−1}x, whence x ∈ t cl A ⊂ [0, γ] · cl A. Therefore (2.3) holds. Moreover, when A is closed, equalities hold in (2.3) and since [0, γ] · A is closed for every γ ∈ [0, ∞[ in view of our convention, we obtain that α_{A} is l.s.c.

(c) The inequalities α_{cl A} ≤ α_{A} ≤ μ_{A} are obvious. It follows that α_{cl A} ≤ \overline{α_{A}} ≤ \overline{μ_{A}}. Let x ∈ X and γ := α_{cl A}(x) ∈ ℝ_{+} ∪ {∞}. If γ = ∞ there is nothing to prove. In the contrary case, by (2.3), there exists t ∈ [0, γ] such that x ∈ t cl A. If t = 0 then x ∈ 0^{+}(cl A) = 0^{+}A. Therefore there exist nets (t_{i}) → 0_{+} and (a_{i}) ⊂ A such that (t_{i}a_{i}) → x. Since μ_{A}(t_{i}a_{i}) ≤ t_{i} for every i, we get that \overline{μ_{A}}(x) ≤ lim inf_{i∈I} μ_{A}(t_{i}a_{i}) ≤ lim inf_{i∈I} t_{i} = 0 ≤ γ. If t ∈]0, ∞[, since t^{−1}x ∈ cl A, there exists a net (a_{i}) ⊂ A converging to t^{−1}x, i.e. (ta_{i}) → x. As above, we obtain that \overline{μ_{A}}(x) ≤ lim inf_{i∈I} μ_{A}(ta_{i}) ≤ t ≤ γ. Therefore α_{cl A} = \overline{μ_{A}}.

(d) Taking (2.1) into account, (2.4) can be reduced to the inclusion μ_{A}^{−1}([0, 1]) ⊂ cl μ_{A}^{−1}([0, 1[). So, let x ∈ μ_{A}^{−1}([0, 1]). Then μ_{A}(\frac{n}{n+1}x) < 1, whence x ∈ cl μ_{A}^{−1}([0, 1[). Finally, in order to prove that \overline{μ_{A}} = μ_{cl A} when A = ]0, 1]A ≠ ∅, since \overline{μ_{A}} = α_{cl A}, it suffices to show that when B is closed and star-shaped one has α_{B} = μ_{B}. These two functions coincide on X \ 0^{+}B. Now, for x ∈ 0^{+}B = ⋂_{t>0} tB one has α_{B}(x) = 0 and μ_{B}(x) = 0 as x ∈ tB for every t > 0.

In the following statement we use the radial tangent cone at 0 to a subset A of X which is the set T^{r}(A, 0) := ⋂_{γ>0}[γ, ∞[A, and the usual tangent cone (or contingent cone) to A at 0, T (A, 0) := ⋂_{γ>0} cl([γ, ∞[A). We note that β_{A}^{−1}(∞) = T^{r}(A, 0) and that when 0 ∉ cl A the cones T^{r}(A, 0) and T (A, 0) are empty. They also coincide when A is a polyhedral convex set or when there exists a neighborhood V of 0 such that A ∩ V is polyhedral.

Proposition 2.4. Let A ⊂ X. Then:

(a) [0, ∞[·A = {x ∈ X | β_{A}^{−1}(x) ≥ 0}, ℙA = {x ∈ X | β_{A}(x) > 0} and 0^{+}A \ ℙA = β_{A}^{−1}({0});

(b) for each γ ∈ [0, ∞[ one has

[γ, ∞[·A ⊂ {x ∈ X | β_{A}(x) ≥ γ} ⊂ [γ, ∞[· cl A.  (2.5)

Moreover, if A is closed and T (A, 0) = T^{r}(A, 0), in particular if 0 ∉ A, then β_{A} = β_{[1,∞[·A} and β_{A} is upper semicontinuous;

(c) β_{cl A} ≥ β_{A} ≥ ν_{A}, \underline{β_{cl A}} = \underline{β_{A}} = \underline{ν_{A}} and for each γ ∈ [0, ∞[ one has

cl ([γ, ∞[·A) = cl (]γ, ∞[·A) = \{x ∈ X | \underline{β_{A}}(x) ≥ γ\}, (2.6)
cl(ℙA) \ ℙ (cl ([1, ∞[A)) = \{x ∈ X | \underline{β_{A}}(x) = 0\}.  (2.7)

Moreover, if T (A, 0) = T^{r}(cl A, 0) then \underline{ν_{A}} = \underline{β_{A}} = β_{cl A}.

Proof. (a) The first two equalities are obvious; they entail the third one.

(b) The first inclusion of relation (2.5) is obvious. For γ = 0 the second inclusion of relation (2.5) follows from (a). Let γ > 0 and let x ∈ X be such that t := β_{A}(x) ≥ γ. Then there exists a sequence (t_{n}) in ]0, ∞[ converging to t and such that x ∈ t_{n}A for each n ∈ ℕ. If t ≠ ∞ we have (t_{n}^{−1}x) → t^{−1}x, whence x ∈ t cl A ⊂ [γ, ∞[· cl A. If t = ∞ then t_{n} ≥ γ for some n ∈ ℕ, and so x ∈ [γ, ∞[·A ⊂ [γ, ∞[· cl A.

Suppose that A = cl A and T (A, 0) = T^{r}(A, 0). Given γ ∈ ℝ_{+}, using the relation

cl ([γ, ∞[·B) = ([γ, ∞[· cl B) ∪ T (B, 0),

valid for any subset B of X, we get that [γ, ∞[·A is closed. Thus, because β_{A} takes its values in {−∞}∪[0, ∞], β_{A} is upper semicontinuous. Moreover, since A ⊂ [1, ∞[·A, β_{A} ≤ β_{[1,∞[·A}. For the converse inequality, consider x ∈ X and γ ∈ ℝ such that β_{[1,∞[·A}(x) > γ.

Then there exists t ≥ max(0, γ) such that x ∈ t ([1, ∞[·A). If t > 0 then x ∈ [t, ∞[·A and thus β_{A}(x) ≥ t ≥ γ. If t = 0 then x ∈ 0^{+} ([1, ∞[·A). Therefore there exist (t_{i}) → 0_{+}, (s_{i}) ⊂ [1, ∞[ and (a_{i}) ⊂ A such that (t_{i}s_{i}a_{i}) → x. Then x ∈ cl ([0, ∞[·A) = [0, ∞[A as observed above. Thus β_{A}(x) ≥ 0 ≥ γ. Therefore β_{A} = β_{[1,∞[·A}.

(c) The inequalities ν_{A} ≤ β_{A} ≤ β_{cl A} are obvious. Therefore \underline{ν_{A}} ≤ \underline{β_{A}} ≤ \underline{β_{cl A}}. Let us show that β_{cl A} ≤ \underline{ν_{A}}. Let x ∈ X and γ ∈ ℝ be such that γ < β_{cl A}(x). There exists t ∈ [max(γ, 0), ∞[ such that x ∈ t cl A. If t = 0 (and so γ ≤ 0), x ∈ 0^{+}(cl A) = 0^{+}A. Therefore there exist nets (t_{i}) → 0_{+} and (a_{i}) ⊂ A such that (t_{i}a_{i}) → x. Since ν_{A}(t_{i}a_{i}) ≥ t_{i} for every i, we get that \underline{ν_{A}}(x) ≥ lim sup ν_{A}(t_{i}a_{i}) ≥ lim sup t_{i} = 0 ≥ γ. The case t ∈]0, ∞[ is similar but simpler. Thus β_{cl A}(x) ≤ \underline{ν_{A}}(x) for every x ∈ X. Therefore \underline{β_{cl A}} = \underline{β_{A}} = \underline{ν_{A}}.

Given γ ∈ ℝ_{+}, the definitions and the inequalities \underline{β_{A}} ≥ β_{A} ≥ ν_{A} yield the inclusions

]γ, ∞[·A ⊂ {x ∈ X | ν_{A}(x) > γ} ⊂ [γ, ∞[·A ⊂ \{x ∈ X | β_{A}(x) ≥ γ\} ⊂ \{x ∈ X | \underline{β_{A}}(x) ≥ γ\}.

Consider x ∈ X such that t := \underline{β_{A}}(x) ≥ γ. By definition, there exists a net (x_{i})_{i∈I} → x such that (β_{A}(x_{i}))_{i∈I} → t. If there exists a cofinal subset J of I such that β_{A}(x_{j}) > γ for each j ∈ J then x_{j} ∈]γ, ∞[A, whence x ∈ cl(]γ, ∞[·A). Thus, we may suppose that γ/2 ≤ β_{A}(x_{i}) ≤ γ for each i ∈ I, and so, t = γ. If γ = 0, then x_{i} ∈ 0^{+}A for each i ∈ I and so x ∈ 0^{+}A; therefore, there exist nets (t_{k})_{k∈K} → 0_{+} and (a_{k})_{k∈K} in A such that (t_{k}a_{k})_{k∈K} → x and so x ∈ cl(]γ, ∞[·A). If γ > 0, consider (ε_{n})_{n∈ℕ} → 0 with ε_{n} ∈]0, γ/2[. Then, for every n ∈ ℕ and i ∈ I, there exists t_{i,n} ∈]β_{A}(x_{i}) − ε_{n}, γ] such that x_{i} ∈ t_{i,n}A. It follows that (t_{i,n}) → γ and x_{i,n} := t_{i,n}^{−1}(γ + ε_{n})x_{i} ∈]γ, ∞[·A. Since (x_{i,n}) → x one has x ∈ cl(]γ, ∞[·A). The proof of relation (2.6) is complete. Taking into account the fact that cl([γ, ∞[·A) = γ cl([1, ∞[·A) for γ > 0, relation (2.7) follows from (2.6). Taking into account the fact that cl([γ, ∞[·A) = γ cl([1, ∞[·A) for γ > 0, relation (2.7) follows from (2.6).

When T^{r}(cl A, 0) = T (A, 0), from (b) we obtain that β_{cl A} is usc and so \underline{ν_{A}} = \underline{β_{A}} = β_{cl A}.

Note that β_{A} and β_{[1,∞[·A} may be different if T (A, 0) ≠ T^{r}(cl A, 0) (f.i. A = {(x, y) ∈ ℝ^{2} | x^{2} + y^{2} ≤ 2y}) or if A is not closed (f.i. A = {(x, y) ∈ ℝ^{2} | x, y > 0, x + y = 1}).

Also note that β_{A} is usc at 0 if and only if 0 ∈ A or 0 ∉ cl A.

Moreover, if X is a normed space and if 0 ∈ cl(A\\{0\}) then \underline{ν_{A}}(0) = +∞. In fact, if (a_{n}) is

a sequence of A\{0} with limit 0, for r_{n} := ∥a_{n}∥^{−1/2} and x_{n} := r_{n}a_{n} we have ν_{A}(x_{n}) ≥ r_{n}, hence \underline{ν_{A}}(0) = +∞. We will see that if one assumes that 0 ∉ cl A the situation is much improved. Some duality observations will be useful for such a purpose and for other aims.

We do not intend to give a complete treatment of the operations with gauges and sets. Let us however consider some important cases, at least for μ and ν.

Proposition 2.5. Let X, Y, Z be vector spaces, A ⊂ X, B ⊂ Y and T : X → Y a linear operator.

(a) For every y ∈ Y one has μ_{T (A)}(y) = inf{μ_{A}(x) | T x = y}.

(b) If 0 ∈ B then for every x ∈ X one has μ_{T^{−1}(B)}(x) = μ_{B}(T x).

(c) If C and D are two starshaped subsets of Z then μ_{C∩D} = μ_{C} ∨ μ_{D}.

(d) If A and B are star-shaped then for all x ∈ X, y ∈ Y one has

μ_{A×B}(x, y) = μ_{A}(x) ∨ μ_{B}(y).  (2.8)

Proof. (a) The conclusion is obvious for y ∉ Im T and for y = 0. Let y ∈ Im T \ {0}. If x ∈ X is such that y = T x, then {t > 0 | x ∈ tA} ⊂ {t > 0 | y ∈ tT (A)}, whence μ_{T (A)}(y) ≤ μ_{A}(x). Hence μ_{T (A)}(y) ≤ inf{μ_{A}(x) | T x = y}. Let now γ ∈ ℝ be such that μ_{T (A)}(y) < γ. Then y ∈ tT (A) = T (tA) for some t ∈]0, γ[. It follows that there exists x ∈ tA with y = T x. Since μ_{A}(x) ≤ t < γ, we obtain that inf{μ_{A}(x) | T x = y} < γ. The conclusion follows.

(b) The proof is simple; just use the definition and the fact that ker T ⊂ T^{−1}(B).

(c) The inequality μ_{C∩D} ≥ μ_{C} ∨ μ_{D} is obvious, while for the reverse one use that C, D are star-shaped.

(d) Apply (c) for Z = X ×Y, C = A×Y, D = X ×B and (b) for the two projections.

As a consequence of the preceding proposition we have the following result about the gauge of a sum which differs from a formula suggested in [7] p. 17. Here we deal with the sublevel convolution (or level convolution or level sum) f◇g of two functions f, g, as used in several recent works (see [3], [17], [26], [29], [30] ...).

Corollary 2.6. Let A, B ⊂ X be star-shaped sets. Then for every x ∈ X,

μ_{A+B}(x) = inf{μ_{A}(u) ∨ μ_{B}(v) | u, v ∈ X, x = u + v} =: (μ_{A}◇μ_{B}) (x).

Proof. Consider T : X × X → X defined by T (u, v) := u + v. Using points (a) and (d) of the preceding proposition, for each x ∈ X we have that

μ_{A+B}(x) = μ_{T (A×B)}(x) = inf{μ_{A×B}(u, v) | u, v ∈ X, u + v = x}
= inf{μ_{A}(u) ∨ μ_{B}(v) | u, v ∈ X, x = u + v}.

Although the conditions are different for ν, the proofs are similar, so that we skip them.

Proposition 2.7. Let X, Y, Z be vector spaces, A ⊂ X, B ⊂ Y and T : X → Y a linear operator.

(a) For every y ∈ Y \ {0} one has ν_{T (A)}(y) = sup{ν_{A}(x) | T x = y}. If A ∩ ker T ⊂ {0}, the preceding formula holds for y = 0, too.

(b) For every x ∈ (X \ ker T) ∪ {0} one has ν_{T^{−1}(B)}(x) = ν_{B}(T x).

(c) If C and D are two shady subsets of Z then ν_{C∩D} = ν_{C} ∧ ν_{D}.

(d) If A and B are shady then for all x ∈ X \ {0}, y ∈ Y \ {0} one has

ν_{A×B}(x, y) = ν_{A}(x) ∧ ν_{B}(y).  (2.9)

Note that, in the framework of the preceding proposition, if x ∈ ker T \{0} then ν_{T^{−1}(B)}(x) = ∞ when 0 ∈ B and ν_{T^{−1}(B)}(x) = −∞ when 0 ∉ B, while sup{ν_{A}(x) | T x = 0} ≥ 1 > 0 = ν_{T (A)}(0) if A ∩ ker T ⊄ {0}.

The next result is a counterpart to Corollary 2.6.

Corollary 2.8. Let A, B ⊂ X be shady sets. Suppose that 0 ∉ A ∪ B ∪ (A + B). Then for every x ∈ ℙ(A + B) ∪ (X \ (ℙA ∪ ℙB)) one has

ν_{A+B}(x) = sup{ν_{A}(u) ∧ ν_{B}(v) | u, v ∈ X, x = u + v}.  (2.10)

In particular, if ℙA ∪ ℙB ⊂ ℙ(A + B), the above formula holds for every x ∈ X.

Proof. Note first that (2.10) holds for x = 0 since 0 ∉ ℙ(A + B) = ℙA + ℙB. So, we may suppose x ∈ X \ {0}. Consider T : X × X → X defined by T (u, v) := u + v. In our hypotheses (A × B) ∩ ker T = ∅. So, from Proposition 2.7 (a) and (d) we have that

ν_{A+B}(x) = ν_{T (A×B)}(x) = sup{ν_{A×B}(u, v) | u, v ∈ X, u + v = x}

= max (sup{ν_{A}(u) ∧ ν_{B}(v) | u, v ∈ X \ {0}, u + v = x}, ν_{A×B}(x, 0), ν_{A×B}(0, x))

= sup{ν_{A}(u) ∧ ν_{B}(v) | u, v ∈ X \ {0}, u + v = x},

(2.11)

since ν_{A×B}(x, 0) = ν_{A×B}(0, x) = −∞, as 0 ∉ A ∪ B.

If x ∈ ℙ(A + B) = ℙA + ℙB, then x = u + v for some u ∈ ℙA, v ∈ ℙB, whence ν_{A}(u) ∧ ν_{B}(v) > 0 ≥ ν_{A}(x) ∧ ν_{B}(0), ν_{A}(0) ∧ ν_{B}(x). Using relation (2.11) we obtain that (2.10) holds in this case. Suppose now that x ∉ ℙA ∪ ℙB. Then ν_{A}(x) ∧ ν_{B}(0) = ν_{A}(0) ∧ ν_{B}(x) = −∞, and so the conclusion follows again using (2.11).

Note that for x ∈ (ℙA ∪ ℙB)\ ℙ(A + B) one has ν_{A+B}(x) = −∞ and sup{ν_{A}(u) ∧ ν_{B}(v) | u, v ∈ X, x = u + v} = 0.

Example 2.9. Consider the following subsets of ℝ^{2} : A_{1} = {(x, y) | y ≥ max(|x| , 1)}, A_{2} = \{(x, y) | y ≥ \sqrt{x^{2} + 1}\} and B = \{(x, y) | x, y > 0, xy ≥ 1\}. Then ℙA_{1} ⊄ ℙ(A_{1} + B) and ℙA_{2} ∪ ℙB ⊂ ℙ(A_{2} + B).

Similar results could be given for the gauges α and β but would involve conditions on the asymptotic cones of images, sums and intersections which are outside the scope of this paper.

3. Harmonic sum and gauges

It is the purpose of the present section to study a variant of the operation of inverse addition of two subsets A, B of a vector space introduced by Rockafellar ([18], [19, p. 21]) as being

A#B := ( ⋃_{λ∈]0,1[} λA ∩ (1 − λ)B) ∪ ({0} ∩ B) ∪ (A ∩ {0}) .

This operation plays an important role in the study of the Plastria’s subdifferential of the sublevel-convolution of two functions which will be undertaken in the paper [16]. We give a direct definition; alternatively, for certain cases, the gauges described above may be used in a way similar to what is done in [24] (in the case of continuous gauges).

For two subsets A, B of X we introduce their harmonic sum, denoted by A◇B, as being A#B, but with the convention made above:

A◇B := ( ⋃_{λ∈]0,1[} λA ∩ (1 − λ)B) ∪ (0^{+}A ∩ B) ∪ (A ∩ 0^{+}B) .

This definition has been introduced by A. Seeger for convex sets containing 0. It also appears as a special case of an operation studied in [21] in the general framework of c^{2}- lattices, at least for closed star-shaped subsets. Note that, if A is a closed cone, then A◇B = A ∩ [0, 1]B. Other examples are described and drawn in [24] or are given below.

Let us note that A◇B is convex when A and B are closed convex subsets. The following result has been obtained by A. Seeger [25], Prop. 4.2 for convex sets containing 0, in which case our assumption ℙA ∩ ℙB ≠ ∅ is automatically satisfied.

Lemma 3.1. Let A, B be two closed subsets of X.

(a) Then the set A◇B is closed and contains A#B.

(b) Moreover, if A, B are convex and ℙA ∩ ℙB ≠ ∅, or if A, B are star-shaped, then

A◇B = cl ( ⋃_{λ∈]0,1[} λA ∩ (1 − λ)B) = cl (A#B) .  (3.1)

Proof. (a) The first assertion follows from the inclusion, valid for any subsets E, F of X,

cl (E#F) ⊂ ( ⋃_{λ∈]0,1[} λ cl E ∩ (1 − λ) cl F ) ∪ (0^{+}E ∩ cl F ) ∪ (cl E ∩ 0^{+}F ) ,

so that the closure of C := ⋃_{λ∈]0,1[} λA∩(1−λ)B is contained in (0^{+}A ∩ B)∪(A ∩ 0^{+}B)∪ C = A◇B. As 0 ∈ 0^{+}D for any subset D of X, the inclusion A#B ⊂ A◇B is immediate.

(b) It is sufficient to show that A◇B ⊂ cl C, or even that 0^{+}A ∩ B ⊂ cl C.

Suppose first that A and B are convex and ℙA ∩ ℙB ≠ ∅, or equivalently, C ≠ ∅, i.e. there exist a ∈ A, b ∈ B and r ∈]0, 1[ such that ra = (1 − r)b. Let x ∈ 0^{+}A ∩ B. Then for each n > 1, for t_{n} := nr(nr − r + 1)^{−1} ∈]0, 1[, we have a_{n} := a + nx ∈ A, b_{n} := (1 − t_{n})b + t_{n}x ∈ B and λ_{n}a_{n} = (1 − λ_{n})b_{n} for λ_{n} := r(nr + 1)^{−1} ∈]0, 1[. Since (λ_{n}a_{n}) → x as n → ∞, we get x ∈ cl C.

Suppose now that A, B are star-shaped. Again, let x ∈ 0^{+}A ∩ B. Since 0^{+}A = ⋂_{t>0} tA, it follows that x_{n} := (1 − \frac{1}{n})x = \frac{1}{n}[(n − 1)x] ∈ \frac{1}{n}A ∩ (1 − \frac{1}{n})B ⊂ C for every n > 1. Since (x_{n}) → x, x ∈ cl C.

Example 3.2. Observe that the relations (3.1) are not valid for non closed sets (even if they are convex and star-shaped); consider the subsets of ℝ^{2}, A = [0, 1] × [0, 1] and B = ] − ∞, 0[×]0, ∞[ ∪{(0, 0)}. Then A#B = {(0, 0)} and A◇B = {0} × [0, 1].

They may fail if A and B are closed non star-shaped subsets; take for example the subsets of ℝ^{2}, A = {(x, x^{2}) | x ∈ ℝ}, B = {0} × [0, ∞[. They may also fail if A and B are closed and convex but ℙA ∩ ℙB is empty. Take for example A = {(x, y) | x > 0, y > 0, xy ≥ 1}, B = [0, ∞[×{0}; then C = ∅, A#B = {0} and A◇B = B.

Lemma 3.3.

(a) For any subsets A, B of X one has 0^{+}(A◇B) ⊂ 0^{+}A ∩ 0^{+}B.

(b) If A and B are closed convex subsets and A◇B ≠ ∅, or if A and B are star-shaped and one of them is closed, then

0^{+}A ∩ 0^{+}B = 0^{+}(A◇B).  (3.2)

Proof. (a) Let x ∈ 0^{+}(A◇B); there exist nets (t_{i})_{i∈I} → 0_{+}, (w_{i})_{i∈I} in A◇B and (λ_{i})_{i∈I} in [0, 1] such that (t_{i}w_{i})_{i∈I} → x and w_{i} ∈ λ_{i}A ∩ (1 − λ_{i})B. If λ_{j} = 0 for j in a cofinal subset of I, then t_{j}w_{j} ∈ 0^{+}A ∩ t_{j}B so that x ∈ 0^{+}A ∩ 0^{+}B. The case λ_{j} = 1 for j in a cofinal subset of I is similar. So, we may suppose that λ_{i} ∈]0, 1[ for every i. Then w_{i} = λ_{i}a_{i} = (1 − λ_{i})b_{i} for some a_{i} ∈ A, b_{i} ∈ B for each i ∈ I; it follows that x = lim_{i} t_{i}λ_{i}a_{i} = lim_{i} t_{i}(1 − λ_{i})b_{i} ∈ 0^{+}A ∩ 0^{+}B.

(b) Suppose A and B are closed convex subsets and A◇B ≠ ∅. Let z_{0} ∈ A◇B and z ∈ 0^{+}A ∩ 0^{+}B. There exists λ ∈ [0, 1] such that z_{0} ∈ λA ∩ (1 − λ)B. Consider first the case λ ∈]0, 1[. Then z_{0} = λx_{0} = (1 − λ)y_{0} for some x_{0} ∈ A and y_{0} ∈ B. It follows that z_{0} + γz = λ(x_{0} + λ^{−1}γz) = (1 − λ)[y_{0} + (1 − λ)^{−1}γz], with x_{0} + λ^{−1}γz ∈ A and y_{0} + (1 − λ)^{−1}γz ∈ B for every γ ≥ 0, whence z_{0} + γz ∈ A◇B for every γ ≥ 0. Therefore z ∈ 0^{+}(A◇B). Suppose now that λ = 0; then z_{0} ∈ 0^{+}A ∩ B. It follows that z_{0} + γz ∈ 0^{+}A ∩ B ⊂ A◇B for every γ ≥ 0, whence, once again, z ∈ 0^{+}(A◇B).

Suppose now that A and B are star-shaped sets, A being closed. In this case 0^{+}A = ⋂_{t>0} tA and 0^{+}B = 0^{+}(cl B) = ⋂_{t>0} t cl B. So, if x ∈ 0^{+}A ∩ 0^{+}B then tx ∈ A ∩ 0^{+}B ⊂ A◇B for every t > 0, whence x ∈ 0^{+}(A◇B).

Example 3.4. Counter-examples for (3.2):

The sets A, B are convex and closed but A◇B = ∅ : A = {(x, y) ∈ ℝ^{2} | x, y > 0, xy ≥ 1}, B = {(x, y) ∈ ℝ^{2} | x < 0, y > 0, xy ≤ −1}; here 0^{+}A ∩ 0^{+}B = {0} × ℝ_{+}.

The sets A, B are closed but are not star-shaped and A◇B ≠ ∅: A = {(x, x^{2}) ∈ ℝ^{2} | x ≥ 0}, B = {(x, x^{2}) ∈ ℝ^{2} | x ≤ 0}; here A◇B = {(0, 0)} and 0^{+}A = 0^{+}B = {0} × ℝ_{+}.

The sets A, B are star-shaped (even convex cones), but none of them is closed: A = {(x, y) ∈ ℝ^{2} | x, y > 0} ∪ {(0, 0)}, B = {(x, y) ∈ ℝ^{2} | x < 0, y > 0} ∪ {(0, 0)}.

Let us note that for A, B ⊂ X,

(]0, 1] · A) # (]0, 1] · B) = ]0, 1] · (A#B), ([1, ∞[·A) # ([1, ∞[·B) = [1, ∞[·(A#B).

In particular, if A, B are star-shaped (shady) then A#B is star-shaped (shady). However, similar relations for the harmonic sum do not hold without additional assumptions. More precisely,

([0, 1] · A) ◇ ([0, 1] · B) = [0, 1] · (A◇B) ⇔ 0^{+}(A◇B) = 0^{+}A ∩ 0^{+}B.

Since for a shady set B we have that 0^{+}B = \overline{cone}B := cl(cone B), we have that 0^{+} ([1, ∞[·A) = \overline{cone}A. Using this relation we obtain easily that

[1, ∞[·(A◇B) ⊂ ([1, ∞[·A) ◇ ([1, ∞[·B) ,

the reverse inclusion being valid if 0^{+}A = \overline{cone}A and 0^{+}B = \overline{cone}B or if A, B are closed and 0 ∉ A ∪ B.

Proposition 3.5. Let A, B ⊂ X. Then μ_{A}+μ_{B} = μ_{A#B} and α_{A}+α_{B} ≤ α_{A◇B}. Moreover,

α_{A} + α_{B} = α_{A◇B} ⇔ 0^{+}(A◇B) = 0^{+}A ∩ 0^{+}B.  (3.3)

Proof. We give the proof for α only; the proof for μ is similar and even simpler.

Let x ∈ X and γ ∈]0, ∞[ be such that α_{A◇B}(x) < γ. Then there exists t ∈ [0, γ] such that x ∈ t · (A◇B). If t = 0 then x ∈ 0^{+}(A◇B). Using Lemma 3.3 we see that α_{A}(x) + α_{B}(x) = 0 ≤ γ. If t > 0 then t^{−1}x ∈ A◇B; therefore t^{−1}x ∈ λA ∩ (1 − λ)B with λ ∈ [0, 1]. So x ∈ λtA ∩ (1 − λ)tB, whence α_{A}(x) + α_{B}(x) ≤ λt + (1 − λ)t = t ≤ γ. It follows that α_{A}(x) + α_{B}(x) ≤ α_{A◇B}(x).

The implication “⇒” of (3.3) follows easily from Proposition 2.3 (a). So suppose that 0^{+}(A◇B) = 0^{+}A ∩ 0^{+}B and take x ∈ X and γ ∈]0, ∞[ such that α_{A}(x) + α_{B}(x) < γ. There exist γ_{1}, γ_{2} ∈ ℝ such that γ = γ_{1} + γ_{2} and α_{A}(x) < γ_{1}, α_{B}(x) < γ_{2}. So there exist t_{1} ∈ [0, γ_{1}], t_{2} ∈ [0, γ_{2}] such that x ∈ t_{1}A ∩ t_{2}B. If t_{1} + t_{2} = 0 then x ∈ 0^{+}A ∩ 0^{+}B, whence, by hypothesis, x ∈ 0^{+}(A◇B); thus α_{A◇B}(x) = 0 ≤ γ. If t_{1} + t_{2} > 0, then

x ∈ (t_{1} + t_{2}) (\frac{t_{1}}{t_{1}+t_{2}} A ∩ \frac{t_{2}}{t_{1}+t_{2}} B) ⊂ (t_{1} + t_{2})(A◇B). Therefore α_{A◇B}(x) ≤ t_{1} + t_{2} ≤ γ. It

follows that α_{A◇B}(x) ≤ α_{A}(x) + α_{B}(x). As the converse inequality is true always, we obtain that α_{A} + α_{B} = α_{A◇B}.

Examples illustrating the way the harmonic sum and gauges operate are given in [24] in the case the gauge μ is continuous; next, we give an example in which this assumption is not satisfied.

Example 3.6. Let A = {(x, y) ∈ ℝ^{2} | x^{2} + y^{2} ≤ 1} and B = {(x, y) ∈ ℝ^{2} | y ≥ x^{2}}. Then

A◇B = \{(x, y) ∈ ℝ^{2} | y ≥ 0, y\sqrt{x^{2} + y^{2}} + x^{2} ≤ y\}.

Indeed, μ_{A}(x, y) = α_{A}(x, y) = \sqrt{x^{2} + y^{2}} and

μ_{B}(x, y) = α_{B}(x, y) = \begin{cases}\frac{x^{2}}{y} & if y > 0, \\ 0 & if x = y = 0, \\ +∞ & if y < 0 or y = 0, x ≠ 0. \end{cases}

Since 0^{+}A ∩ 0^{+}B ⊂ 0^{+}A = {0}, it follows from Propositions 2.3 and 3.5 that

A◇B = cl(A#B) = {(x, y) ∈ ℝ^{2} | α_{A}(x, y) + α_{B}(x, y) ≤ 1}
= \{(x, y) ∈ ℝ^{2} | y ≥ 0, y\sqrt{x^{2} + y^{2}} + x^{2} ≤ y\}.

We observe that, since 0 is in the interior int A of A, A#B ⊂ int A, but the point (0, 1) of A◇B belongs to the boundary of A.

A similar result holds for ν and β.

Proposition 3.7. Let A, B ⊂ X. Then, with the convention that (−∞) + (+∞) = −∞, one has ν_{A} + ν_{B} = ν_{A#B} and β_{A◇B} ≤ β_{A} + β_{B}. Moreover,

β_{A} + β_{B} = β_{A◇B} ⇔ 0^{+}A ∩ 0^{+}B ⊂ [0, ∞[·(A◇B).  (3.4)

In particular β_{A} + β_{B} = β_{A◇B} whenever 0^{+}A ∩ 0^{+}B = 0^{+}(A◇B).

If A and B are closed, shady and do not contain 0, then 0^{+}(A◇B) = [0, ∞[·(A◇B), and

β_{A} + β_{B} = β_{A◇B} ⇔ 0^{+}(A◇B) = 0^{+}A ∩ 0^{+}B.

Proof. Again, the proof for ν is similar to the proof for β. Let x ∈ X and γ ∈ ℝ be such that β_{A◇B}(x) > γ. Then there exists t ≥ max(0, γ) such that x ∈ t · (A◇B). If t = 0 then x ∈ 0^{+}(A◇B). Since 0^{+}(A◇B) ⊂ 0^{+}A ∩ 0^{+}B, we obtain that β_{A}(x), β_{B}(x) ≥ 0, and so β_{A}(x) + β_{B}(x) ≥ 0 ≥ γ. Consider now the case t > 0. Then t^{−1}x ∈ A◇B; therefore t^{−1}x ∈ λA ∩ (1 − λ)B with λ ∈ [0, 1]. So x ∈ λtA ∩ (1 − λ)tB, whence β_{A}(x) + β_{B}(x) ≥ λt + (1 − λ)t = t ≥ γ. It follows that β_{A◇B}(x) ≤ β_{A}(x) + β_{B}(x).

Suppose that β_{A}+β_{B} = β_{A◇B} and take x ∈ 0^{+}A∩0^{+}B. Then β_{A◇B}(x) = β_{A}(x)+β_{B}(x) ≥ 0, whence, by Proposition 2.4, x ∈ [0, ∞[·(A◇B).

The proof of the implication “⇐” of (3.4) is similar to the corresponding one in (3.3), and so is omitted.

Assume now that A and B are shady; then A◇B is also shady. As for a closed shady set C not containing 0 one has 0^{+}C = cl(ℙC) = ℝ_{+}C, the conclusion follows.

4. Harmonic sum and polarities

In the sequel we assume that the space X is an Hausdorff locally convex space and X^{∗} is its topological dual. Throughout, for a function f, ∂f and f^{∗} denote the subdifferential and the conjugate of f in the sense of convex analysis.

Let us consider the polar and anti-polar of A ⊂ X defined by

A^{∘} := {x^{∗} ∈ X^{∗} | ⟨x, x^{∗}⟩ ≤ 1 ∀ x ∈ A},
A^{∇} := {x^{∗} ∈ X^{∗} | ⟨x, x^{∗}⟩ ≥ 1 ∀ x ∈ A},

with A^{∘} = A^{∇} = X^{∗} if A = ∅. Here we use the notation of [1] and [2]; for other uses of the anti-polar set, see [4], [9], [11], [27], [28]. We will use the following obvious relations:

A^{∘} = (\overline{co}(A ∪ \{0\})^{∘} = (\overline{co}(]0, 1]A))^{∘} , A^{∇} = (\overline{co}([1, ∞[A))^{∇} .

Let us note that since A^{∘} is closed, convex and contains 0, one has

0^{+}A^{∘} = A^{−} := {x^{∗} ∈ X^{∗} | ⟨x, x^{∗}⟩ ≤ 0 ∀ x ∈ A},

while if A^{∇} ≠ ∅

0^{+}A^{∇} = A^{+} := {x^{∗} ∈ X^{∗} | ⟨x, x^{∗}⟩ ≥ 0 ∀ x ∈ A}.

Taking the last relation into consideration, in this section we consider that 0 · A^{∇} = A^{+}, even if A^{∇} = ∅. Note that if A is non-empty and convex, then by an easy separation argument one has A^{∇} = ∅ if and only if 0 ∈ cl A (see also [4]).

In the sequel we denote by ι_{C} the indicator function of a subset C of X given by ι_{C}(x) = 0 if x ∈ C, +∞ else. In the following lemma we use the well-known fact that the conjugate h^{∗} of a positively homogeneous function h : X → ℝ ∪ {+∞} is the indicator function of H := {x^{∗} ∈ X^{∗} | x^{∗} ≤ h} and, when h(0) = 0, H = ∂h(0), the subdifferential of h at 0. The relation (μ_{A})^{∗} = ι_{A^{∘}} of the next lemma is given in [6] Proposition 1.6 and in [19]. It implies that the support function of A^{∘} is (μ_{A})^{∗∗}; when A is convex and contains 0, this last function is \overline{μ_{A}} and one gets connections with section 6 of [25] which is devoted to support functions of direct and inverse additions of convex sets.

Lemma 4.1. Let A ⊂ X be a non-empty set. Then (μ_{A})^{∗} = (α_{A})^{∗} = ι_{A^{∘}}, (−ν_{A})^{∗} = (−β_{A})^{∗} = ι_{−A^{∇}} and

A^{∘} = ∂μ_{A}(0) = ∂α_{A}(0),  (4.1)
A^{∇} = −∂(−ν_{A})(0) = \{x^{∗} ∈ X^{∗} | x^{∗} ≥ β_{A}\} = \{x^{∗} ∈ X^{∗} | x^{∗} ≥ \underline{β_{A}}\}.  (4.2)

In particular, if A is convex, then \underline{β_{A}}(0) = 0 if and only if 0 ∉ cl A; in such a case one has ∂(−\underline{β_{A}})(0) = −A^{∇}.

Proof. We prove the assertions dealing with β_{A} and ν_{A}, the proof of those concerning α_{A} and μ_{A} being similar. Moreover, the equalities (−ν_{A})^{∗} = (−β_{A})^{∗} = ι_{−A^{∇}} are consequences of (4.2) and of the observation preceding the statement. Given x^{∗} ∈ A^{∇} let us prove that ⟨x, x^{∗}⟩ ≥ ν_{A}(x) for every x. We may suppose x ≠ 0. Let λ < ν_{A}(x); there exists t > max(λ, 0) such that t^{−1}x ∈ A, whence ⟨t^{−1}x, x^{∗}⟩ ≥ 1, i.e. ⟨x, x^{∗}⟩ ≥ t ≥ λ. Hence ⟨x, x^{∗}⟩ ≥ ν_{A}(x). Conversely, suppose that ⟨x, x^{∗}⟩ ≥ ν_{A}(x) for every x ∈ X. If x ∈ A, ⟨x, x^{∗}⟩ ≥ ν_{A}(x) ≥ 1. Hence x^{∗} ∈ A^{∇}. Thus the first equality in (4.2) holds. The other ones follow from \underline{ν_{A}} = \underline{β_{A}} (by Proposition 2.4) and a passage to the limit superior.

Suppose now that A is convex. If 0 ∉ cl A, then A^{∇} is nonempty, whence, from (4.2) \underline{β_{A}}(0) ≤ 0. Therefore, \underline{β_{A}}(0) = 0. Conversely, if \underline{β_{A}}(0) = 0 then 0 ∉ cl A and β_{A} is usc at 0, with β_{A}(0) = 0. The relation ∂(−\underline{β_{A}})(0) = −A^{∇} then follows from (4.2).

Lemma 4.2. Let A ⊂ X be a non-empty convex set. Then A^{∘∘} = \overline{co}(A∪\{0\}). If 0 ∉ cl A then (A^{∇})^{+} = \overline{cone}A = (0^{+}A^{∇})^{+} and A^{∇∇} = cl([1, ∞[·A); if 0 ∈ cl A then A^{∇∇} = X.

Proof. The first relation is the well-known bipolar theorem.

Suppose that 0 ∉ cl A; as mentioned above, A^{∇} ≠ ∅. Of course, using also Proposition 2.4, we have that

A ⊂ A^{∇∇} ⊂ (A^{∇})^{+} = (\overline{cone}A^{∇})^{+} = (0^{+}A^{∇})^{+} .

Therefore \overline{cone}A ⊂ (A^{∇})^{+}. Let x_{0} ∉ \overline{cone}A. By a separation theorem, there exists x_{0}^{∗} ∈ X^{∗} such that ⟨x_{0}, x_{0}^{∗}⟩ < 0 ≤ ⟨x, x_{0}^{∗}⟩ for every x ∈ A. Take x_{1}^{∗} ∈ A^{∇}; then for every λ ≥ 0 we have that ⟨x, x_{1}^{∗} + λx_{0}^{∗}⟩ ≥ 1 for every x ∈ A, whence x_{1}^{∗} + λx_{0}^{∗} ∈ A^{∇} for every λ ≥ 0. As ⟨x_{0}, x_{1}^{∗} + λx_{0}^{∗}⟩ < 0 for sufficiently great λ, we obtain that x_{0} ∉ (A^{∇})^{+}.

The last part can be found in [4].

We are ready to deal with the harmonic sum of two sets. The formula given below eases the proof of the following well-known result (which can be deduced from Proposition 2 of [32] or from a separation theorem).

Recall ([32]) that if f, g : X → \overline{ℝ} are proper lower semi-continuous sublinear functions, then

∂(f + g)(0) = cl (∂f(0) + ∂g(0)) ,  (4.3)

the closure being taken with respect to the weak^{∗} topology.

Note that the above formula may not be valid if one of the functions is not lsc; just take f = ι_{P} and g = ι_{Q}, where P = ℝ_{+}^{2} and Q = {(0, 0)} ∪ {(x, y) ∈ ℝ^{2} | x < 0, y > 0}.

Taking into account that 0^{+}A^{∘} = A^{−}, A^{∘} ⊂ (0^{+}A)^{∘} and 0^{+}A^{∇} = A^{+}, it is easy to obtain the following inclusions for any non-empty subsets:

A^{∘}◇B^{∘} ⊂ (A + B)^{∘},  cl(A^{∘} + B^{∘}) ⊂ (A◇B)^{∘},  (4.4)
A^{∇}◇B^{∇} ⊂ (A + B)^{∇},  cl(A^{∇} + B^{∇}) ⊂ (A◇B)^{∇}.  (4.5)

The following proposition completes [25] Theorem 6.1; the second part of its assertion (a) is given in [6] Theorem 1.2^{0}. There the harmonic sum is even considered for an arbitrary finite number of convex subsets containing 0. Here we only consider two sets; however, given a family A_{1}, ..., A_{n} of subsets of X one can set

A_{1}◇ ⋯ ◇A_{n} := ⋃_{s∈S} (s_{1}A_{1} ∩ ⋯ ∩ s_{n}A_{n})

where S is the canonical simplex of ℝ^{n}, i.e. the set of s := (s_{1}, ..., s_{n}) ∈ ℝ_{+}^{n} such that s_{1} + ⋯ + s_{n} = 1. When A_{1}, ..., A_{n} are closed convex subsets of X containing 0 (or, more generally, if ℙA_{1} ∩ ⋯ ∩ ℙA_{n} ≠ ∅), this set coincides with ((A_{1}◇A_{2})◇ ⋯) ◇A_{n}.

Proposition 4.3. Let A, B be two non-empty closed convex subsets of X.

(a) If A◇B ≠ ∅ then (A◇B)^{∘} = cl(A^{∘} + B^{∘}). If 0 ∈ A ∩ B then A^{∘}◇B^{∘} = (A + B)^{∘}.

(b) If A◇B ≠ ∅ then (A◇B)^{∇} = cl(A^{∇} + B^{∇}).

(c) If A and B are shady then A^{∇}◇B^{∇} = (A + B)^{∇}.

Proof. (a) A direct proof can be given along the lines of the proof of (c) presented below. As a short variant, use Lemma 4.1, Propositions 2.3, 3.5 and relations (4.3), (4.4) to get

(A◇B)^{∘} = ∂α_{A◇B}(0) = ∂ (α_{A} + α_{B}) (0) = cl (∂α_{A}(0) + ∂α_{B}(0)) = cl (A^{∘} + B^{∘}) .

The second formula follows from the above one applied for A^{∘} and B^{∘}, and the bipolar theorem.

(b) If 0 ∈ A then 0 ∈ 1A∩0B ⊂ A◇B and A^{∇} = ∅. It follows that (A◇B)^{∇} = ∅ = cl(A^{∇}+ B^{∇}). Thus we may assume that 0 ∉ A ∪ B; then the sublinear functions −β_{A} and −β_{B} are lsc and take the value 0 at 0 by Proposition 2.4; moreover, (−β_{A}) + (−β_{B}) = −β_{A◇B} by Proposition 3.7 and Lemma 3.3. Thus −β_{A◇B}(0) = 0 and by Lemma 4.1

−(A◇B)^{∘} = ∂ (−β_{A◇B}) (0) = cl (∂ (−β_{A}) (0) + ∂ (−β_{B}) (0)) = cl (−A^{∇} − B^{∇}) .

(c) Let z^{∗} ∈ (A + B)^{∇}; thus ⟨x, z^{∗}⟩ + ⟨y, z^{∗}⟩ ≥ 1 for all x ∈ A, y ∈ B. Since A = [1, ∞[·A and B = [1, ∞[·B, we have that r := inf_{x∈A} ⟨x, z^{∗}⟩ ≥ 0, s := inf_{y∈B} ⟨y, z^{∗}⟩ ≥ 0 and r + s ≥ 1. Suppose first that r, s > 0; there exist r′, s′ > 0 such that r ≥ r′, s ≥ s′ and r′ + s′ = 1. It follows that z^{∗} ∈ r′A^{∇} ∩ s′B^{∇} ⊂ A^{∇}◇B^{∇}. Suppose now that r = 0. Then z^{∗} ∈ A^{+} ∩ B^{∇} = 0 · A^{∇} ∩ 1 · B^{∇} ⊂ A^{∇}◇B^{∇}. Using relation (4.5), we obtain that A^{∇}◇B^{∇} = (A + B)^{∇}.

Example 4.4. Note that the formulae (A◇B)^{∘} = cl(A^{∘} + B^{∘}), (A◇B)^{∇} = cl(A^{∇} + B^{∇}) may not hold when A◇B = ∅. Take for instance A = −B = [1, ∞[×ℝ. Then A◇B = ∅, A^{∘} = −B^{∘} =]−∞, 0]×{0} and A^{∇} = −B^{∇} = [1, ∞[×{0} so that A^{∘}+B^{∘} = A^{∇}+B^{∇} = ℝ × {0}.

5. Computation of the flat and sharp conjugates of a function

A number of conjugacies adapted to quasiconvex functions are known. In most cases the computation of the conjugate of a function is not obvious. It is the purpose of this section to present some cases for which an explicit expression of the conjugate can be given.

We start with the flat and the sharp conjugacies which have proved to be useful, especially for Hamilton-Jacobi equations (see [5], [31], [10]). They are defined on X^{∗} × ℝ by

f^{♯}(x^{∗}, s) := sup {⟨x, x^{∗}⟩ | x ∈ X, f(x) ≤ s} ,
f^{♭}(x^{∗}, s) := sup {⟨x, x^{∗}⟩ | x ∈ X, f(x) < s} .

Note that f^{♯}(x^{∗}, s) = f^{♭}(x^{∗}, s) if s > inf f and f is convex because in this case {x ∈ X | f(x) ≤ s} ⊂ cl{x ∈ X | f(x) < s}; this equality also holds for many quasiconvex functions.

Let us first give a slight extension of [31] Proposition 15 which deals with the flat conjugate of a function of the form f := φ ∘ ∥·∥, where φ : ℝ_{+} → \overline{ℝ} is nondecreasing. Here we consider the case of a function of the form f := φ ∘ g, where φ : \overline{ℝ} → \overline{ℝ} is nondecreasing and g : X → \overline{ℝ} is sublinear.

As in [31] we use the lowest quasi-inverse φ^{e} and the greatest quasi-inverse φ^{h} of φ introduced in [13] by

φ^{e}(s) := sup \{r ∈ \overline{ℝ} | φ(r) < s\} = inf \{t ∈ \overline{ℝ} | s ≤ φ(t)\} ,
φ^{h}(s) := sup \{r ∈ \overline{ℝ} | φ(r) ≤ s\} = inf \{t ∈ \overline{ℝ} | s < φ(t)\} ,

which are characterized, respectively, by the implications

φ(r) < s ⇒ r ≤ φ^{e}(s) and t < φ^{e}(s) ⇒ φ(t) < s,
φ(r) ≤ s ⇒ r ≤ φ^{h}(s) and t < φ^{h}(s) ⇒ φ(t) ≤ s.

In order to obtain formulae for (φ ∘ g)^{♭} and (φ ∘ g)^{♯} with φ nondecreasing and g sublinear we need the following result which completes Theorem 13.5 of [19] in asserting that the infimum is attained when finite in the formula (5.1) below.

Lemma 5.1. Let g : X → \overline{ℝ} be a proper convex function, γ > inf g and x^{∗} ∈ X^{∗}. Then,

g^{♭}(x^{∗}, γ) = g^{♯}(x^{∗}, γ) = min{λγ + h(x^{∗}, λ) | λ ≥ 0},  (5.1)

where

h(x^{∗}, λ) = (λg)^{∗}(x^{∗}) = \begin{cases}λg^{∗}(λ^{−1}x^{∗}) & if λ > 0, \\ sup\{⟨x, x^{∗}⟩ | x ∈ dom g\} & if λ = 0. \end{cases}

Proof. Consider

F : X × ℝ → \overline{ℝ}, F (x, t) = \begin{cases}− ⟨x, x^{∗}⟩ & if g(x) ≤ γ + t, \\ +∞ & otherwise. \end{cases}

It is obvious that F is a convex function and P_{ℝ}(dom F) ⊃] inf g − γ, ∞[, where P_{ℝ} is the projection from X × ℝ onto ℝ. Therefore 0 ∈ int P_{ℝ}(dom F). Applying the fundamental duality theorem in the case of a finite dimensional parameter space (see for example [33, Th. 2.6.5]) we have that inf_{x∈X} F (x, 0) = max_{λ∈ℝ} (−F^{∗}(0, λ)), in the sense that the supremum is attained when it is finite, or equivalently,

sup{⟨x, x^{∗}⟩ | g(x) ≤ γ} = min_{λ∈ℝ}F^{∗}(0, −λ).

But

F^{∗}(0, −λ) = sup{−λt + ⟨x, x^{∗}⟩ | x ∈ X, t ∈ ℝ, g(x) ≤ t + γ}
= λγ + sup{−λr + ⟨x, x^{∗}⟩ | (x, r) ∈ epi g}.

It is obvious that F^{∗}(0, −λ) = ∞ if λ < 0 and F^{∗}(0, −λ) = sup{⟨x, x^{∗}⟩ | x ∈ dom g} if λ = 0. Let λ > 0; then

F^{∗}(0, −λ) = λγ + sup{⟨x, x^{∗}⟩ − λg(x) | x ∈ dom g} = λγ + λg^{∗}(λ^{−1}x^{∗}).

The conclusion follows.

When g is sublinear, we obtain formulae involving the gauges studied above which will be used in our paper [16].

Proposition 5.2. Let g : X → \overline{ℝ} be a proper sublinear functional, lsc at 0, and let x^{∗} ∈ X^{∗}. Then for every γ > inf g one has

g^{♭}(x^{∗}, γ) = g^{♯}(x^{∗}, γ) = \begin{cases}γ · α_{∂g(0)}(x^{∗}) & if γ > 0, \\ ι_{[0,∞[∂g(0)}(x^{∗}) & if γ = 0, \\ γ · β_{∂g(0)}(x^{∗}) & if γ < 0, \end{cases}

with α_{∂g(0)}(x^{∗}) and β_{∂g(0)}(x^{∗}) attained when finite.

Note that here ∂g(0) is nonempty, as g is lsc at 0, and 0^{+}(∂g(0)) = (dom g)^{−} .

Proof. Recall that, since g is sublinear, dom g is a convex cone and g^{∗} = ι_{∂g(0)}. In our conditions the function h introduced in Lemma 5.1 is given by

h(x^{∗}, λ) = \begin{cases}0 & if λ > 0, x^{∗} ∈ λ∂g(0) or λ = 0, x^{∗} ∈ (dom g)^{−}, \\ +∞ & otherwise, \end{cases}

and so

{x^{∗} ∈ X^{∗} | ∃λ ≥ 0, h(x^{∗}, λ) < ∞} = [0, ∞[·∂g(0).  (5.2)

Let γ > 0. It is obvious that g^{♯}(x^{∗}, γ) ≥ 0 in this case. From relations (5.1) and (5.2), we have that g^{♯}(x^{∗}, γ) = α_{∂g(0)}(x^{∗}) = ∞ if x^{∗} ∉ [0, ∞[·∂g(0). If x^{∗} ∈ (dom g)^{−} then h(x^{∗}, 0) = 0, and so, from (5.1), g^{♯}(x^{∗}, γ) = 0 = γα_{∂g(0)}(x^{∗}). Suppose now that x^{∗} ∈ ]0, ∞[·∂g(0) \ (dom g)^{−}. Applying again the preceding lemma we have that g^{♯}(x^{∗}, γ) < ∞ so that

g^{♯}(x^{∗}, γ) = min\{λγ + h(x^{∗}, λ) | λ ≥ 0\} = \overline{λ}γ + h(x^{∗}, \overline{λ})

for some \overline{λ} ≥ 0. Since x^{∗} ∉ (dom g)^{−}, \overline{λ} ≠ 0, and so \overline{λ} > 0 and x^{∗} ∈ \overline{λ} · ∂g(0). It follows that

g^{♯}(x^{∗}, γ) = min\{λγ | λ ≥ 0, x^{∗} ∈ λ∂g(0)\} = \overline{λ}γ = γα_{∂g(0)}(x^{∗}).

Suppose now that inf g < 0.

For γ = 0, from (5.1) and (5.2), we have that g^{♯}(x^{∗}, 0) = ι_{[0,∞[∂g(0)}(x^{∗}).

Let γ < 0. As above, if x^{∗} ∉ [0, ∞[·∂g(0) then g^{♯}(x^{∗}, γ) = γ · ν_{∂g(0)}(x^{∗}) = ∞. If x^{∗} ∈]0, ∞[·∂g(0) then, from (5.1), g^{♯}(x^{∗}, γ) < 0, and so

g^{♯}(x^{∗}, γ) = min{λγ | λ ≥ 0, x^{∗} ∈ λ∂g(0)} = γ max{λ ≥ 0 | x^{∗} ∈ λ∂g(0)}
= γ · β_{∂g(0)}(x^{∗}) < 0

with β_{∂g(0)}(x^{∗}) attained. If x^{∗} ∈ (dom g)^{−}\]0, ∞[·∂g(0) from (5.1) and (5.2) we get that g^{♯}(x^{∗}, γ) = 0. The proof is complete.

The previous result can be extended to functions of the form φ ∘ g with φ nondecreasing and g sublinear. It has been obtained by Volle [31] when g is a norm on X.

Proposition 5.3. Let φ : \overline{ℝ} → \overline{ℝ} be nondecreasing and g : X → \overline{ℝ} be a proper convex function. Consider f := φ ∘ g and (x^{∗}, r) ∈ X^{∗} × ℝ. If φ^{e}(r) ∈] inf g, ∞[ then

f^{♭}(x^{∗}, r) = g^{♭} (x^{∗}, φ^{e}(r)) = g^{♯} (x^{∗}, φ^{e}(r)) ,

and for φ^{h}(r) ∈] inf g, ∞[ one has

f^{♯}(x^{∗}, r) = g^{♯} (x^{∗}, φ^{h}(r)) = g^{♭} (x^{∗}, φ^{h}(r)) .

Moreover, when g is sublinear and lsc at 0, and φ^{e}(r) ∈] inf g, ∞[,

f^{♭}(x^{∗}, r) = \begin{cases}φ^{e}(r) · α_{∂g(0)}(x^{∗}) & if φ^{e}(r) > 0, \\ ι_{[0,∞[∂g(0)}(x^{∗}) & if φ^{e}(r) = 0, \\ φ^{e}(r) · β_{∂g(0)}(x^{∗}) & if φ^{e}(r) < 0, \end{cases}

similar expressions being valid for f^{♯}(x^{∗}, r) with φ^{e}(r) replaced by φ^{h}(r) ∈] inf g, ∞[.

Proof. For the first assertions it is sufficient to observe that

{x | g(x) < φ^{e}(r)} ⊂ {x | f(x) < r} ⊂ {x | g(x) ≤ φ^{e}(r)} ⊂ cl{x | g(x) < φ^{e}(r)},
{x | g(x) < φ^{h}(r)} ⊂ {x | f(x) ≤ r} ⊂ {x | g(x) ≤ φ^{h}(r)} ⊂ cl{x | g(x) < φ^{h}(r)},

while for the last ones it is enough to apply the preceding proposition.

Using Lemma 5.1 we also obtain the next formula for the normal cone to a sublevel set of a convex function; note that such a formula is usually stated for finite convex functions. Recall that the normal cone to a subset C of X at x_{0} ∈ X is given by

N(C, x_{0}) := {x^{∗} ∈ X^{∗} | ⟨x^{∗}, x − x_{0}⟩ ≤ 0 ∀ x ∈ C} .

Proposition 5.4. Let g : X → \overline{ℝ} be a proper convex function and x_{0} ∈ dom g. If g(x_{0}) > inf g then

N([g ≤ g(x_{0})], x_{0}) = [0, ∞[·∂g(x_{0}),

with the convention that 0 · ∂g(x_{0}) = 0^{+}∂g(x_{0}) = N(dom g, x_{0}).

Proof. The inclusion ⊃ is obvious. Let x^{∗} ∈ N([g ≤ g(x_{0})], x_{0}) be fixed. Using the definition of the normal cone and Lemma 5.1 with γ = g(x_{0}) we have

⟨x_{0}, x^{∗}⟩ = sup{⟨x, x^{∗}⟩ | x ∈ [g ≤ γ]} = λg(x_{0}) + h(x^{∗}, λ)

for some λ ≥ 0. If λ = 0 then ⟨x_{0}, x^{∗}⟩ = sup{⟨x, x^{∗}⟩ | x ∈ dom g}, and so x^{∗} ∈ N(dom g, x_{0}) = 0 · ∂g(x_{0}). If λ > 0 then ⟨x_{0}, x^{∗}⟩ = λg(x_{0}) + λg^{∗}(λ^{−1}x^{∗}), whence λ^{−1}x^{∗} ∈ ∂g(x_{0}). The proof is complete.

Remark 5.5. The relation 0 · ∂g(x_{0}) = N(dom g, x_{0}) is justified by the fact that 0^{+}∂g(x_{0}) = N(dom g, x_{0}) when ∂g(x_{0}) ≠ ∅. Moreover, when ℙ(dom g − x_{0}) = X one has N(dom g, x_{0}) = {0}.

Example 5.6. Let g : ℝ → \overline{ℝ} be given by g(x) = −\sqrt{1 − x^{2}} for |x| ≤ 1, +∞ otherwise. Taking x_{0} = 1, we observe that ∂g(x_{0}) = ∅ but N([g ≤ g(x_{0})], x_{0}) = N(dom g, x_{0}) = 0 · ∂g(x_{0}).

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