Samir Adly
email: samir.adly@unilim.fr

Florent Nacry
email: florent.nacry@univ-perp.fr

Lionel Thibault
email: lionel.thibault@umontpellier.fr

Prox-regular sets and Legendre-Fenchel transform related to separation properties

Keywords: Prox-regular set, strongly convex set, Legendre-Fenchel transform, semiconvexity, quadratic support functional, geometric separation

This paper is devoted to nonconvex/prox-regular separations of sets in Hilbert spaces. We introduce the Legendre-Fenchel r-conjugate of a prescribed function and r-quadratic support functionals and points of a given set, all associated to a positive constant r. By means of these concepts we obtain nonlinear functional separations for points and prox-regular sets. In addition to such functional separations, we also establish geometric separation results with balls for a prox-regular set and a strongly convex set.

Introduction

The present paper is concerned with the interest of the use of the Legendre-Fenchel transform in the study of certain other crucial features for prox-regular sets which allow us to provide fundamental separation results for such sets. By means of the usual Legendre-Fenchel transform ( [START_REF] Legendre | Mémoire sur l'intégration de quelques équations aux différences partielles[END_REF][START_REF] Fenchel | On conjugate convex functions[END_REF]), we define the concept of r-conjugate f ,r of a function f : X → R ∪ {-∞, +∞} as the Legendre-Fenchel conjugate of f + 1 2r • 2 , where X is a Hilbert space and r ∈]0, +∞] is a positive extended-real number. We develop the study of r-conjugates and we obtain diverse remarkable properties, especially for semiconvex functions. A fixed point result is also obtained for the transform f → f ,r . Considering the fundamental case when f is the indicator function of a subset S of X gives rise to the quadratic functional q x * ,r := x * , • -1 2r • 2 on X for every x * ∈ X. Through this, the usual notions of support linear functionals and support points in functional analysis are extended to support quadratic functionals and quadratically supported points. We study in details the properties of these new notions mainly for (uniformly) r-prox-regular sets in X. We show in particular a Bishop-Phelps ( [START_REF] Bishop | The support functionals of convex sets[END_REF]) type result for the density of rquadratically supported points of the set S in its boundary whenever S is r-prox-regular. With the above concepts and their properties at hands, we establish functional separation results for points outside prox-regular sets and also ball separation results (that is, separation with balls) between an r-prox-regular set and an s-strongly convex set (see Section 5 for definition).

The paper is outlined as follows. Section 2 recalls the notions and results useful for our analysis. Clarke tangent cones, normal cones and subdifferentials as well as proximal normals and proximal subdifferentials, as basic tools in our approach, are recalled in Subsection 2.1. The definition and main results needed in the paper for prox-regular sets are given in Subsection 2.2. The concepts of r-conjugate, quadratic support functional and quadratically supported point are defined and analyzed in Section 4. Proposition 3.1 in this section shows properties of quadratic support functionals and points, Proposition 3.4 illustrates particular features of f ,r when f is semiconvex, and Proposition 3.5 demonstrates the fixed point result for the transform f → f ,r . In Section 4, for any r-prox-regular set in X we establish in Proposition 4.1 and Theorem 4.3, functional separation type results for points at positive distances less than the radius r of proxregularity. Ball separation type results between prox-regular sets and strongly convex sets are provided in Theorem 5.3 and Theorem 5.4 in the last Section 5.

Notation and preliminaries

As usual, N denotes the set of integers starting from 1 and R := R ∪ {-∞, +∞} is the extended real-line. Throughout the paper, X stands for a (real) Hilbert space not reduced to the trivial space {0} endowed with the inner product •, • and its associated norm

• := •,
• . The open (resp. closed) ball and the sphere of X centered at x ∈ X with radius ρ > 0 are denoted by B(x, ρ) (resp. B[x, ρ]) and S(x, ρ). In the particular case of the closed unit ball and the unit sphere of X centered at zero, we use the following notation B := B[0 X , 1] and S := S(0 X , 1). Now, consider any nonempty subset S of X. The distance function d S from S is defined as d S (x) = d(x, S) := inf y∈S xy for all x ∈ S.

The multimapping Proj S : X ⇒ X of nearest points in S is defined by Proj S (x) = Proj(S, x) := {y ∈ S : xy = d S (x)} for all x ∈ X.

Whenever the latter set is reduced to a singleton for some x ∈ X, that is Proj S (x) = {y}, the vector y ∈ S is denoted by proj S (x) or P S (x). Let

Tangent and normal cones

Throughout this paragraph, S is a subset of X which contains a vector x.

The Bouligand tangent cone T B (S; x) of S at x ∈ S is defined as the set of h ∈ X such that there exist a sequence (t n ) n∈N of positive reals with t n → 0 and a sequence (h n ) n∈N of X with h n → h such that

x + t n h n ∈ S for all n ∈ N.
The Clarke tangent cone T C (S; x) of S at x ∈ S is the set of h ∈ X such that for every sequence (x n ) n∈N of S with x n → x, for every sequence (t n ) n∈N of positive reals with t n → 0, there is a sequence (h n ) n∈N of X with h n → h satisfying

x n + t n h n ∈ S for all n ∈ N; the negative polar cone of T C (S; x) is the Clarke normal cone N C (S; x) of S at x, i.e, N C (S; x) = {ζ ∈ X : ζ, h ≤ 0, ∀h ∈ T C (S; x)}.
Given a nonempty open set U in X and a function f : U → R, the Clarke subdifferential ∂ C f (x) of f at a point x ∈ U where f is finite is defined by

∂ C f (x) := {ζ ∈ X : (ζ, -1) ∈ N C epi f ; (x, f (x)) }, where epi f := {(u, r) ∈ U ×R : f (u) ≤ r} is the epigraph of f . By convention, ∂ C f (x) = ∅ when f (x) is not finite. Defining the indicator function ψ S of the set S by ψ S (x ) := 0 if x ∈ S, +∞ if x ∈ X \ S, one has ∂ C ψ S (x ) = N C (S; x ) for all x ∈ S.
Besides the above notions of Clarke normal cone and subdifferential, let us introduce the proximal normal ones. The proximal normal cone of S at x ∈ S, denoted by N P (S; x), is defined as (see, e.g., [START_REF] Rockafellar | Variational Analysis[END_REF])

N P (S; x) := {v ∈ X : ∃r > 0, x ∈ Proj S (x + rv)} .

By convention, one puts

T C (S; x ) = ∅ and N C (S; x ) = N P (S; x ) = ∅ for all x ∈ X \ S. It is not difficult to see that v ∈ N P (S; x) if and only if there is a real r > 0 such that v, x -x ≤ 1 2r x -x 2 for all x ∈ S; (2.1)
in such a case we will say that v is a proximal normal to S at x with constant r > 0. It is known and not difficult to see that

N P (S; x) ⊂ N C (S; x).
According to the definition of the proximal normal cone, we notice that, for any v ∈ X such that Proj S (v) = ∅, we have vw ∈ N P (S; w) for all w ∈ Proj S (v).

(2.2)

Coming back to the above function f : U → R defined on the nonempty open set U of X, a vector ζ ∈ X is said to be a proximal subgradient of f at a point x ∈ U with f (x) finite, provided there are a real σ ≥ 0 and a real η > 0 with B(x, η) ⊂ U such that

ζ, y -x ≤ f (y) -f (x) + σ y -x 2 for all y ∈ B(x, η),
which is known to be equivalent to (ζ, -1) ∈ N P epi f ; (x, f (x)) . The set ∂ P f (x) of all proximal subgradients of f at x is the proximal subdifferential of f at x. Like for the Clarke subdifferential, one sets ∂ P f (x) = ∅ whenever f is not finite at x.

If S is convex, it is known (and easily seen) that the two normal cones N P (S; x) and N C (S; x) coincide with the normal cone in the sense of convex analysis, that is,

N P (S; x) = N C (S; x) = {ζ ∈ X : ζ, y -x ≤ 0, ∀y ∈ S} .
(2.3)

If f : U → X is a convex function defined on a nonempty open convex subset U of X, then the two above subdifferentials coincide on U with the subdifferential in the sense of convex analysis, i.e., for every x ∈ U ,

∂ P f (x) = ∂ C f (x) = {ζ ∈ X : ζ, y -x ≤ f (y) -f (x), ∀y ∈ U } =: ∂f (x).

Prox-regular sets in the Hilbert setting

This succinct paragraph is devoted to the needed basic features on prox-regularity which is at the heart of the present paper. Prox-regular sets (a term coined in [START_REF] Poliquin | Local differentiability of distance functions[END_REF]) has a long history which goes back to the famous work [START_REF] Federer | Curvature measures[END_REF] by H. Federer with the class of positively reached sets in R n . Over the years, various names have been introduced in different contexts to refer to such sets: weakly convex ( [START_REF] Vial | Strong and weak convexity of sets and functions[END_REF]) or proximally smooth sets ( [START_REF] Clarke | Proximal smoothness and the lower-C 2 property[END_REF]) are commonly used in Hilbert setting; for other names we refer to the work [START_REF] Colombo | Prox-regular sets and applications[END_REF]. Studies of prox-regular sets in Banach spaces can be found in [START_REF] Bernard | Zlateva Characterizations of prox-regular sets in uniformly convex Banach spaces[END_REF][START_REF] Bernard | Prox-regular sets and epigraphs in uniformly convex Banach spaces: various regularities and other properties[END_REF][START_REF] Balashov | Weakly convex and proximally smooth sets in Banach spaces[END_REF]. Besides those theoretical developments, let us mention that prox-regular sets have also been sucessfully involved in many concrete problems as the celebrated Moreau's sweeping process (see, e.g., the survey [START_REF] Nacry | Regularization of sweeping process: old and new[END_REF] and the references therein). For more details, in both theoretical and concrete approaches, one can refer for instance to [START_REF] Poliquin | Local differentiability of distance functions[END_REF], the survey [START_REF] Colombo | Prox-regular sets and applications[END_REF] and the book [START_REF] Thibault | Unilateral Variational Analysis in Banach Spaces[END_REF] along with the references therein. We also refer to [START_REF] Adly | Preservation of Prox-Regularity of Sets with Applications to Constrained Optimization[END_REF][START_REF] Adly | Prox-regularity approach to generalized equations and image projection[END_REF][START_REF] Venel | A numerical scheme for a class of sweeping processes[END_REF] for various stability properties.

Definition 2.1 Let S be a nonempty closed subset of X and r ∈]0, +∞]. One says that S is r-prox-regular (or uniformly prox-regular with constant r) whenever, for every x ∈ S, for every v ∈ N P (S; x) ∩ B and for every real t ∈]0, r], one has

x ∈ Proj S (x + tv).

Given a closed subset S ⊂ X, x ∈ S and v ∈ N P (S; x) with v = 1, it is known and easily seen that for every real t > 0 one has

x ∈ Proj S (x + tv) ⇔ S ∩ B(x + tv, t) = ∅.
(2.4)

In such a case, one says that the unit normal proximal vector v to S at x is realized by the t-ball B(x + tv, t).

The following theorem provides some useful characterizations and properties of uniform prox-regular sets for which we refer to [START_REF] Poliquin | Local differentiability of distance functions[END_REF][START_REF] Colombo | Prox-regular sets and applications[END_REF][START_REF] Thibault | Unilateral Variational Analysis in Banach Spaces[END_REF]. (c) On the open set U r (S) the mapping proj S is well-defined, and for every real s ∈]0, r[, for all x, x ∈ U s (S),

proj S (x) -proj S (x ) ≤ 1 1 -(s/r) x -x .
(d) For any u ∈ Tube r (S) such that proj S (u) exists, one has

proj S (u) = proj S proj S (u) + t u -proj S (u) u -proj S (u) for all t ∈ [0, r[. (e) The function d 2 S is C 1,1 on U r (S)
and ∇d 2 S (x) = 2 xproj S (x) for all x ∈ U r (S). (f ) The set S is tangentially regular as well as normally regular in the sense that T B (S; x) = T C (S; x) and N P (S; x) = N C (S; x) for all x ∈ X.

Further, one has

∂ P d S (x) = ∂ C d S (x) for all x ∈ U r (S).
Then, the assertions (a), (b), (c), (d), (e) are pairwise equivalent and each one implies the assertion (f ).

When S is r-prox-regular, according to the above assertion (f) we will just write N (S; •) in place of any of N P (S; •) and N C (S; •), except when we need to precise the type of normal cone we are working with. If C is ∞-prox-regular in X (i.e., nonempty closed and convex) then N (C; •) coincides with the normal cone in the sense of convex analysis, that is,

N (C; x) = {ζ ∈ X : ζ, y -x ≤ 0, ∀y ∈ C} for all x ∈ C.
Remark 2.1 In addition to (d) in the above theorem, it is worth noticing that for an r-prox-regular subset of X with r ∈]0, +∞[, we have the crucial inclusion proj S (u) ∈ Proj S proj S (u) + r uproj S (u) uproj S (u) for all u ∈ Tube r (S).

C ⊂ { x , • ≤ σ(x , C)} and x , x ≥ σ(x , C).
It should be noted that the latter inequality is in fact an equality, that is

x , x = σ(x , C).

(3.1)

Coming back to an r-prox-regular set S of X for some real r > 0, let us take a look at the meaning of the non-nullity similar property N P (S; x) = {0} for some x ∈ bdry S.

Fix any x ∈ N P (S; x) \ {0} with x = 1. According to the r-prox-regularity property (see Theorem 2.1), we have

x , u -x ≤ 1 2r u -x 2 for all u ∈ S,
or equivalently,

x + x r , u - u 2 2r ≤ x + x r , x - x 2 2r
for all u ∈ S. Defining the quadratic functional q y * ,r : X → R by q y ,r (x) := y , xx 2 2r for all x ∈ X, we observe in the same line as (3.1) that q y ,r (x) = ϕ S,r (y ).

This leads to introduce the following definition.

Definition 3.1 Let S be a nonempty closed subset of X and r ∈]0, +∞] be an extended real. Given u ∈ X, we say that q u ,r is an r-quadratic support functional of S whenever there exists x ∈ S such that

q u ,r (x) := u , x - 1 2r x 2 = sup x∈S ( u , x - 1 2r x 2 ) = sup x∈S q u ,r (x) =: ϕ S,r (u ).
In such a case, u is said to quadratically support S at x. We say that x ∈ S is an r-quadratic support point of S whenever there exists u ∈ X with u = x r such that q u ,r supports S at x. The set of all r-quadratic support points of S is denoted by supp r (S).

Whenever r = +∞, the above definition is reduced to saying that u * , • (resp. x) is a support linear functional (resp. support point) of S in the usual sense of functional analysis. Recall that u , • with u ∈ X is a support linear functional to a nonempty closed subset S of X provided there is some x ∈ S such that

u , x = sup x∈S u , x =: σ(u , S),
and if u = 0, such a point x ∈ S is said to be a support point of S. If in addition S is convex, it is readily seen that the latter equality is equivalent to the following inclusion u ∈ N (S; x).

Through the next proposition, we see that the notions of r-quadratic support functionals and points are related to the proximal normal cone of S. Proposition 3.1 Let S be a closed subset of X, u ∈ X, x ∈ S, r ∈]0, +∞]. The following hold. (a) The functional q u ,r is an r-quadratic support functional of S if and only if there exists

z ∈ S such that u - z r , x -z ≤ 1 2r x -z 2 for all x ∈ S.
(a ) The vector x is an r-quadratic support point of S supported by q u ,r if and only if

u = x r and u - x r , x -x ≤ 1 2r x -x 2 for all x ∈ S.
(b) If q u ,r is an r-quadratic support functional of S, then there exists z ∈ S such that uz r is a proximal normal to S at z with constant r. (b ) If x is an r-quadratic support point of S supported by the quadratic functional q u ,r , then there exists a nonzero proximal normal v to S at x with constant r such that

u = v + x r .
(c) If q u ,r is an r-quadratic support functional of S, then for every real α > 0, q αu ,r/α is an r α -quadratic support functional of S. (c ) If x is an r-quadratic support point of S supported by the quadratic functional q u ,r , then x is an r α -quadratic support point of S supported by the quadratic functional q αu ,r/α .

If in addition the set S is r-prox-regular, one has: (d) q u ,r is an r-quadratic support functional of S whenever there exists z ∈ S such that

u ∈ N (S; z) ∩ B + z r .
(d ) The vector x is an r-quadratic support point of S supported by the r-quadratic functional q u ,r whenever u = x r and u ∈ N (S; x) ∩ B + x r .

(e) One has the equality

supp r (S) = {x ∈ bdry X S : N (S; x) = {0}} .
Proof. (a) and (a ): To establish (a), it suffices to fix any z ∈ S and to note that for every x ∈ S,

u , x - 1 2r x 2 ≤ u , z - 1 2r z 2 ⇔ u , x -z ≤ 1 2r x 2 -z 2 ⇔ u , x -z ≤ 1 2r x -z 2 + 2 x -z, z ⇔ u - z r , x -z ≤ 1 2r x -z 2 .
The equivalence provided by

u - z r , x -z ≤ 1 2r x -z 2 for all x ∈ S.
Applying (a) yields that q u ,r is an r-quadratic support functional of S.

(e) The inclusion supp r (S) ⊂ {x ∈ bdry X S :

N (S; x) = {0}}
is a direct consequence of (b ). In order to establish the converse inclusion, fix any x ∈ bdry X S such that N (S; x) = {0}. Let any v ∈ N (S; x) with v = 1. We have

u := v + x r ∈ N (S; x) ∩ B + x r and u = x r (otherwise v = 0). It follows from (d ) that
x is an r-quadratic support point of the set S.

Remark 3.1 Consider any nonempty closed set S of X with S = X. According to [27, Theorem 2.10 and Proposition 2.6], we know that the set

x ∈ bdry S : N F (S; x) = {0}
is dense in bdry S, where N F (S; •) denotes the Fréchet normal cone of S (see, e.g., [START_REF] Mordukhovich | Variational Analysis and Generalized Differentiation I[END_REF]). Taking into account the property (e) in the above proposition and assuming in addition that S is r-prox-regular for some r ∈]0, +∞], we arrive to the density of supp r (S) in the boundary of S. This can be seen as an prox-regular extension of the celebrated Bishop-Phelps theorem (see [START_REF] Bishop | The support functionals of convex sets[END_REF]).

Let S be an r-prox-regular subset of X and x ∈ Tube r (S). Thanks to the inclusion

R + (x -proj S (x)) ⊂ N (S; proj S (x)) we have x -proj S (x) d S (x) + proj S (x) r ∈ N (S; proj S (x)) ∩ B + proj S (x) r . Denoting Γ S,r (x) := ( 1 r - 1 d S (x) )proj S (x) + x d S (x) , (3.2)
we see that q Γ S,r (x),r -quadratically supports S at proj S (x) according to (d) in Proposition 3.1 above. Since x ∈ Tube r (S), we must have Γ S,r (x) = proj S (x) r (otherwise, x = proj S (x) which is a contradiction), and then (d ) above says that proj S (x) is an r-quadratic support point of S quadratically supported by q Γ S,r (x),r .

We summarize those features in the following proposition.

Proposition 3.2 Let S be an r-prox-regular set of X with r ∈]0, +∞] and x ∈ Tube r (S).

The following hold: (a) q Γ S,r (x),r -quadratically supports S at proj S (x); (b) proj S (x) is an r-quadratic support point of S supported by q Γ S,r (x),r .

The quantity in Definition 3.1

ϕ S,r (x ) := sup x∈S x , x - x 2 2r
can be seen as a particular Legendre-Fenchel conjugate since

ϕ S,r = ψ S + • 2 2r .
Let us point out here that ϕ S,1 (which is sometimes called Asplund function) is greatly involved in convex analysis and approximation theory (see, e.g., [START_REF] Asplund | Cebysev sets in Hilbert space[END_REF][START_REF] Berens | Best approximation in Hilbert space[END_REF][START_REF] Hiriart-Urruty | Ensembles de Tchebychev vs ensembles convexes : l'état de la situation vu via l'analyse convexe non lisse[END_REF][START_REF] Hiriart-Urruty | La conjecture des points les plus éloignés revisités[END_REF]). Various and numerous properties of ϕ S,r are given in our paper [START_REF] Adly | New metric properties for prox-regular sets[END_REF]. In particular, it is shown that such a function is the right tool to extend fundamental metric properties known in Convex Analysis to the framework of Variational Analysis of prox-regular sets. The important role of ϕ S,r leads us to introduce the following definition for any function.

Definition 3.2 Given a function f : X → R and an extended real r ∈]0, +∞], one defines the Legendre-Fenchel r-conjugate f ,r :

X → R ∪ {-∞, +∞} by f ,r (x ) := f + 1 2r • 2 (x ) = sup x∈X x , x - 1 2r x 2 -f (x) for all x ∈ X. (3.3)
The following proposition states a list of properties which can be verified in a straightforward way. Proposition 3.3 Let f : X → R be a function and r > 0 be an extended real. The following hold. (a) One has the equality f ,r (0) = -inf

X (f + 1 2r • 2 ). (b) One has the equivalences -∞ ∈ f ,r (X) ⇔ f ≡ +∞ ⇔ f ,r ≡ -∞. (c) If the function f ,r is proper, then f is proper. (d) For every (x , t) ∈ X × R, one has the equivalence (x , t) ∈ epi f ,r ⇔ x , • - 1 2r • 2 -t ≤ f (•).
(e) The function f has no continuous r-quadratic minorants if and only if f ,r ≡ +∞. (f ) If dom f ,r = ∅, then f is bounded from below on each bounded subset of X.

(g) The function f ,r is convex and lower semicontinuous.

(h) For all x ∈ X, one has

f ,r (x ) = sup x∈dom f ( x , x - 1 2r x 2 -f (x)) = sup (x,t)∈epi f ( x , x - 1 2r x 2 -t).
(i) If f is a proper function, then one has the following Fenchel-Young inequality

f (x) + f ,r (x ) ≥ x , x - 1 2r x 2 for all x , x ∈ X.
The following example will be crucial in order to get fixed points of for the r-conjugate transforms f → f ,r with r > 0 (see Proposition 3.5).

Example 1 Let S be a nonempty subset of X and let r > 0 be a positive real. Fix any real α > -1 2r and set

f := ψ S + α • 2 .
From the very definition of r-conjugate , we see with

ρ := r 2αr+1 > 0 f ,r (x ) = ϕ S,ρ (x ) = ρ 2 x 2 - 1 2ρ d 2 S (ρx ) for all x ∈ X. (3.4)
In particular, with S = X we obtain

(α • 2 ) ,r = r 2(2rα + 1) • 2 . (3.5)
As usual, a function f : X → R is said to be proper if its effective domain domf := {x ∈ X : f (x) < +∞} is nonempty (i.e., f ≡ +∞) along with f (X) ⊂ R ∪ {+∞}. Recall that if f is proper then the celebrated Fenchel-Moreau theorem (see, e.g., [START_REF] Borwein | Convex Functions: constructions, characterizations and counterexamples[END_REF][START_REF] Moreau | Fonctionnelles Convexes[END_REF][START_REF] Thibault | Unilateral Variational Analysis in Banach Spaces[END_REF]) asserts that f is lower semicontinuous and convex if and only if f = f * * .

In the statement of the next proposition, the concepts of infimal convolution and semiconvexity will be involved. Taking two functions f, g : X → R ∪ {+∞}, we recall (see [START_REF] Moreau | Fonctionnelles Convexes[END_REF]) that the infimal convolution f g : X → R of f and g is defined by

(f g)(x) := inf y∈X f (x -y) + g(y) for all x ∈ X.
The following definition concerns the second concept of semiconvexity (see,e.g., [START_REF] Cannarsa | Semiconcave Functions, Hamilton-Jacobi Equations, and Optimal Control[END_REF]).

Definition 3.3 A function f : C → R ∪ {+∞} defined on a nonempty convex subset C of X is said to be σ-semiconvex (on C) for some σ ∈ R + := [0, +∞[ if f (tx + (1 -t)y) ≤ tf (x) + (1 -t)f (y) + σ 2 t(1 -t) x -y 2 ,
for all x, y ∈ C and for all t ∈]0, 1[,

or equivalently if f + σ 2 • 2 is convex on C.
The function f is said to be semiconvex near a point if it is semiconvex on a ball centered at that point.

Recall that if f is σ-semiconvex near a point x, then for some real η > 0

∂ P f (x) = ∂ C f (x) = ζ ∈ H : ζ, h ≤ f (x + h) -f (x) + σ 2 h 2 ∀h ∈ B(x, η) .
Related to the semiconvexity property we also have.

Proposition 3.4 Let f : X → R be a function and r > 0 be an extended real.

(j) If f is r -1 -semiconvex and lower semicontinuous on X with f (X) ⊂ R ∪ {+∞}, then one has f (x) = sup

x ∈X q x ,r (x) -f ,r (x ) for all x ∈ X.

(k) If f is proper, then f is r -1 -semiconvex and lower semicontinuous on X if and only

if f = (f ,r ) -1 2r • 2 . (l) If f is proper, r -1
-semiconvex and lower semicontinuous on X, then one has

(x, u - 1 r x) ∈ gph ∂ C f ⇔ (u, x) ∈ gph ∂f ,r . (m) (f ,r ) = (f + 1 2r • 2 ) . (n) (f ,r ) = f + 1 2r • 2 if and only if f is proper, r -1 -semiconvex and lower semicon- tinuous. (o) (f ) ,r = (f r 2 • 2 ) . (p) (f ,r ) = (f r 2 • 2 ) if and only if (f r 2 •
2 ) is proper, convex and lower semicontinuous.

Proof. (j) Under the assumption in (j) the function g(•) := f (•) + 1 2r • 2 is proper, lower semicontinuous and convex. Fix any x ∈ X. According to the Fenchel-Moreau theorem, we may write

g(x) = sup x ∈X ( x , x -g (x )),
or equivalently

f (x) + 1 2r x 2 = sup x ∈X ( x , x -(f + 1 2r • 2 ) (x )).
Consequently, we derive

f (x) = sup x ∈X ( x , x - 1 2r x 2 -f ,r (x ))
which is the desired equality. According to (3.3), we can write

(f ,r ) = [(f + 1 2r • 2 ) ] = (f + 1 2r • 2 ) ,
which is (m). Invoking again (3.3), we have

(f ) ,r = (f + 1 2r • 2 ) = (f + g ) ,
where g(•) := r 2 • 2 . It remains to apply the equality f + g = (f g) (see, e.g., [28, (6.15), p. 45], [START_REF] Thibault | Unilateral Variational Analysis in Banach Spaces[END_REF]) to get the desired equality in (o). This finishes the proof of the proposition.

We establish now the fixed point result for the r-conjugate transform. Proposition 3.5 Given a function f : X → R and a real r > 0, one has

f ,r (•) = f (•) ⇔ f (•) = -1 + √ 1 + 4r 2 4r • 2 .
Proof. Fix any real r > 0. Let us consider the function g : R → R defined by

g(t) := 4rt 2 + 2t -r for all t ∈ R.
It is a routine to observe that g has exactly two zeros, namely

α := -1 + √ 1 + 4r 2 4r ≥ 0 and β := -1 - √ 1 + 4r 2 4r . ⇐, Assume that f (•) = α • 2 .
Observe first that the equality g(α) = 0 entails ρ 2 = α with ρ := r 2αr+1 . Then, using the formulae (3.4) provided by Example 1 furnishes

f ,r (•) = ρ 2 • 2 = α • 2 = f (•). ⇒, Assume that f ,r (•) = f (•). According to Proposition 3.3(b), -∞ / ∈ f ,r (X) = f (X) (otherwise f = f ,r
) and then f ≡ +∞. Thus, f is a proper function and this allows us to apply the extended Fenchel-Young inequality in Proposition 3.4(i) above to obtain

f (x) ≥ 2r -1 4r x 2 for all x ∈ X. (3.6)
Now, let us consider the function h :

I :=] -1 2r , +∞[→ R given by h(t) := r 2 1 2rt + 1 for all t ∈ I.
Note that α is the only one fixed point for h. Further, it is clear that h is decreasing on I along with h(I) ⊂ I. Hence, with u 0 := 2r-1 4r ∈ I, we can set

u n := h(u n-1
) for all n ∈ N.

Rewriting (3.6) as u 0 • 2 ≤ f , we see through the equality (3.5

) that f = f ,r ≤ h(u 0 ) • 2 = u 1 • 2 , so u 0 • 2 ≤ f ≤ u 1 • 2
. By induction we derive that

u 2n x 2 ≤ f (x) ≤ u 2n+1 x 2 for all n ∈ N, x ∈ X. (3.7)
Observe that for all t ∈ I,

(h • h)(t) -t = - g(t) 2r 2 + 4rt + 2 .
Thanks to the inclusion u 0 ∈]β, α[∩I, we have g(u 0 ) ≤ 0, and hence

0 ≤ (h • h)(u 0 ) -u 0 = u 2 -u 0 .
Consequently, the sequence (u 2n ) n≥0 is nondecreasing, and then (see the definition of (u n ) n≥0 ) we have the fact that (u 2n+1 ) n≥0 is nonincreasing. Coming back to (3.7), we get that (u 2n ) n≥0 (resp. (u 2n+1 ) n≥0 ) is bounded from above (resp. below). We deduce that u 2n → α and u 2n+1 → α as n → +∞. It remains to let n → ∞ in (3.7) to complete the proof of the proposition.

Remark 3.2 Let S be an r-prox-regular subset of X with r > 0. Let also C be any nonempty closed convex subset of X with C ⊂ U s (S) for some real 0 < s < r. It is known (see, e.g., [START_REF] Balashov | Weak convexity of the distance function[END_REF][START_REF] Thibault | Unilateral Variational Analysis in Banach Spaces[END_REF]) that the distance function d S (•) is (rs) -1 -semiconvex on C, or equivalently d S + ψ C is semiconvex on X. This along with Proposition 3.4(j) allows us to write

(d S + ψ C )(x) = sup x ∈X (q x ,r-s (x) -(d S + ψ C ) ,r-s (x )) for all x ∈ X.
Consequently, for any x ∈ C we have

d S (x) = sup x ∈X q x ,r-s (x) -sup y∈X x , y - y 2 2(r -s) -(d S + ψ C )(y) = sup x ∈X q x ,r-s (x) -sup y∈C x , y - y 2 2(r -s) -d S (y) = sup
x ∈X inf y∈C q x ,r-s (x) -q x ,r-s (y) + d S (y) .

Now, assume for a moment that the nonempty closed set S is convex and take any x ∈ X. The latter equalities with C := X give

d S (x) = sup x ∈X inf y∈X x , x -y + d S (y) . Writting inf y∈X x , x -y + d S (y) = inf u∈S x , x -u -sup y∈X [ x * , y -u -y -u ] = inf u∈S x * , x -u -ψ B X (x * ) ,
we then obtain the known formula (see, e.g., [18, Remarks 7.2,p.126])

d S (x) = sup x ∈B X inf y∈S x , x -y = max x ∈B X inf y∈S x , x -y ,
where the latter equality is due to the weak upper semicontinuity of x → x , xy . We mention here that an extension of the latter inf-sup formula to the context of prox-regular sets is established in our paper [START_REF] Adly | New metric properties for prox-regular sets[END_REF].

Prox-regularity and functional separation

It has been well-recognized that the geometric Hahn-Banach theorems are among of the most important and powerful principles of functional analysis (see, e.g., [START_REF] Fabian | The Basis for Linear and Nonlinear Analysis[END_REF][START_REF] Deutsch | Best Approximations in Inner Product Spaces[END_REF][START_REF] Mordukhovich | Variational Analysis and Generalized Differentiation I[END_REF][START_REF] Rockafellar | Variational Analysis[END_REF] and the references therein). Roughly speaking, the geometric Hahn-Banach theorem for closed convex sets asserts that a compact convex set A and a closed convex set B of X (or more generally of a locally convex space) with A ∩ B = ∅ can be separated by a hyperplane/half-space. The case where A is reduced to a singleton (so A is an exterior point of B) is of a great interest and can be stated as follows (see also [START_REF] Deutsch | Best Approximations in Inner Product Spaces[END_REF]Theorem 6.23]).

Theorem 4.1 Let S be a nonempty closed convex set of X, x ∈ X \ S. Then, one has with x := d S (x) -1 (xproj S (x)) ∈ S the following separation property for some α ∈ R, Applying the latter theorem with S := B -A for two subsets A, B ⊂ X and x := 0 leads to the following general convex separation result: Theorem 4.2 Let A and B be two nonempty disjoint closed and convex sets in X with A + B closed (which holds if B is weakly compact). Then, there exist x ∈ X and a real α > 0 such that sup

S ⊂ { x , • < α} ⊂ H ≤ (x , α) := { x , • ≤ α}.
a∈A x , a + α ≤ inf b∈B x , b , in particular σ(x , A) ≤ -σ(x , -B) = -σ(-x , B).
It is worth noticing that, to argue the two above results, Zorn lemma is not needed because of the Hilbertian structure of X.

Our aim in this section is to extend such separation results to the context of r-proxregular sets. As expected, the (classical) support function σ S (•) = ϕ S,∞ (•) (resp. the linear form q x ,∞ (•) = x , • ) will be replaced by ϕ S,r (•) (resp. the quadratic function q x ,r (•)). Let us mention that going beyond convexity in separation theory is a very challenging issue which has been (apparently) developed in a very few number of works. General nonconvex separation lies at the heart of the construction of Mordukhovich's generalized differentiation with the so-called extremal variational principle (see, e.g., [27, Chapter 2] and the references therein). To the best of our knowledge, the first separation result for prox-regular sets is due to J.P. Vial. In his paper, Vial shows ([37, Theorem 5.1]) in the finite dimensional setting that a weakly convex set of constant r (a concept equivalent to the r-prox-regularity in Hilbert spaces) and an R-strongly convex set can always be separated by a ball whenever a certain condition holds between the radii r, R.

Vial also provides an estimate for the radius of the involved ball, depending on r and R. Such a result has been successfuly extended to the Hilbert framework by G.E. Ivanov [START_REF] Ivanov | Weakly Convex Sets and Functions: Theory and Applications[END_REF] (see also the survey [START_REF] Goncharov | Strong and weak convexity of closed sets in a Hilbert space[END_REF]) and by M.V. Balashov and G.E. Ivanov ([6]) with separation result in the context of uniformly convex and smooth Banach spaces. In this section and the next one, we provide other additional separation properties.

Let us start by extending Theorem 4.1 to the context of prox-regular sets. The following easy result will be needed. Lemma 4.1 Let S be a nonempty closed subset of X, x ∈ X \ S. Assume that p ∈ Proj S (x) = ∅ and let ρ be an extended real such that ρ > d S (x). The following hold with x := d S (x) -1 (px). (a) One has the estimate

x , x + 1 ρ d 2 S (x) < x , x + d S (x) = x , p . (4.1)
(b) If in addition the set S is r-prox-regular for some r ∈]0, +∞], then for all y ∈ S,

x , x + 1 ρ d 2 S (x) < x , p ≤ x , y + 1 2r p -y 2 .
Proof. The assertion (a) is a direct consequence of the inequality ρ -1 d S (x) < 1. Let us show (b). Assume that S is r-prox-regular for some extended real r > 0 and fix any y ∈ S.

Noting that -x ∈ N P (S; p) (see (2.2)), we can apply Theorem 2.1 to get

-x , y -p ≤ x 2r y -p 2 ,
or equivalently

x , p ≤ x , y + 1 2r y -p 2 . (4.2)
Combining (4.1) and (4.2), we arrive to

x , x + d 2 S (x) ρ < x , p ≤ x , y + 1 2r p -y 2 .
The proof is complete.

We derive from the particular case when S is r-prox-regular and r ≥ ρ the following separation type result. Proposition 4.1 Let S be an r-prox-regular subset of X with r ∈]0, +∞]. Let x ∈ Tube r (S) and x := d S (x) -1 proj S (x) -x , and let ρ > d S (x). Then, for all y ∈ S x , x +

d 2 S (x) ρ < x , proj S (x) ≤ x , y + 1 2r proj S (x) -y 2 ,
so there exists α ∈ R such that for all y ∈ S,

x , x + d 2 S (x) ρ < α < x , y + 1 2r proj S (x) -y 2 .
Remark 4.1 In addition to the above result, diverse intrinsic metric characterizations of prox-regular sets can be found in [START_REF] Balashov | Weak convexity of the distance function[END_REF][START_REF] Bounkhel | Nonconvex sweeping process and prox-regularity in Hilbert space[END_REF][START_REF] Nacry | Distance function from a prox-regular set[END_REF][START_REF] Poliquin | Local differentiability of distance functions[END_REF][START_REF] Thibault | Unilateral Variational Analysis in Banach Spaces[END_REF].

In Theorem 4.1, the separation property holds thanks to the vector x = d S (x) -1 (xproj S (x)) ∈ X \ {0}. In the context of r-prox-regular sets, the vector x will be replaced by Γ S,r (x x) x (see (3.2)). The next lemma shows in particular that Γ S,r (x) ∈ X \ 1 r S. For x, y ∈ X, we denote as usual ]x, y[:= {(1 -t)x + ty : t ∈]0, 1[}.

) := ( 1 r -1 d S (x) )proj S (x)+ 1 d S (
y = λ + (1 -λ)(1 - r d ) p + (1 -λ) r d u = 1 -(1 -λ) r d p + (1 -λ) r d u, hence d S (y) = d S 1 -(1 -λ) r d p + (1 -λ) r d u ≤ (1 -λ) r d u -p = (1 -λ) r d d < r,
where the first inequality is due to p ∈ S. By contradiction, suppose that y ∈ S. Then, obviously proj S (y) is well defined and satisfies proj S (y) = y. On the other hand, the inclusions y ∈]p, rΓ S,r (u)[ and p ∈ Proj S rΓ S,r (u) give proj S (y) = p. It follows that y = p and this is the desired contradiction. (d) Fix any y ∈]proj S (x), rΓ S,r (x)[. According to (c) above, we have y ∈ Tube r (S), in particular Γ S,r (y) is well defined. Further, the second inclusion provided by (b) above guarantees that proj S (y) = proj S (x) =: p. Through elementary computations, we see that

ω := Γ S,r (x) -Γ S,r (y) = ( 1 d S (y) - 1 d S (x) )(p -x) + 1 d S (y) (x -y),
hence

ω 2 = (d S (x) -d S (y)) 2 d 2 S (y) + x -y 2 d 2 S (y) + 2 ( 1 d S (y) - 1 d S (x) )(p -x), x -y d S (y) . (4.3)
Now, let us distinguish two cases: Case 1: y ∈]p, x[. In such a case, we have

p -x = d S (x) x -y (y -x) and d S (x) = d S (y) + x -y . (4.4)
Then, it is not difficult to check that

( 1 d S (y) - 1 d S (x) )(p -x), x -y d S (y) = - x -y 2 d 2 S (y) . (4.5)
Putting together (4.3), (4.4) and (4.5), we arrive to

ω 2 = x -y 2 d 2 S (y) + x -y 2 d 2 S (y) -2 x -y 2 d 2 S (y) = 0, i.e., Γ S,r (y) = Γ S,r (x). Case 2: y ∈]x, rΓ S,r (x)[. It is clear that p -x = d S (x) x -y (x -y) and d S (y) = d S (x) + x -y .
Proceeding as in Case 1, we obtain ω = 0, which is the desired equality. (e) Fix any s ∈]0, r[. Observe first that we obviously have with p x := proj S (x) and p y := proj S (y)

{(1 - r s )p y + r s y : y ∈ D s (S)} ⊂ Λ r (S) := {(1 - r d S (x) )p x + r s x : x ∈ Tube s (S)}.
Let us show the converse inclusion. Let x ∈ Tube r (S). Set v x := rΓ S,r (x). Thanks to the continuity of θ : [0, 1] → R + defined by

θ(t) := d S p x + t(v x -p x ) for all t ∈ [0, 1],
we can choose some t 0 ∈ [0, 1] such that

y 0 := p x + t 0 (v x -p x ) ∈ D s (S).
It remains to apply (d) above to get v x = rΓ S,r (y 0 ). (f ) In view of (c), there is only one inclusion to justify. Fix any x ∈ Tube r (S). By (a) above, we know that x ∈]proj S (x), rΓ S,r (x)[. Defining the function θ : [0, 1] → R by θ(t) := d S proj S (x) + t rΓ S,r (x) -proj S (x) for all t ∈ [0, 1] and noticing by (b) above that θ(1) = r, the intermediate value theorem gives some

y ∈ D s (S)∩]proj S (x), rΓ S,r (x)[.
We derive from (d) that Γ S,r (y) = Γ S,r (x). On the other hand, we have proj S (x) = proj S (y) because Proj S (rΓ S,r (x)) proj S (x). The inclusion x ∈]proj S (x), rΓ S,r (x)[ in (a) can then be rewritten as

x ∈]proj S (y), rΓ S,r (y)[.

The proof is complete.

With the above results at hands, we are ready to establish a separation theorem by means of Γ S,r (•). In order to state the theorem, let us denote for any x ∈ X, ρ, α ∈ R

Q x ,ρ,α := {q x ,ρ ≤ α} := x ∈ X : x , x - x 2 2ρ ≤ α .
Theorem 4.3 Let S be an r-prox-regular subset of X with r ∈]0, +∞], x ∈ Tube r (S).

Then, one has with x) x the following separation property for some α ∈ R

x := Γ S,r (x) = ( 1 r -1 d S (x) )proj S (x) + 1 d S (
S ⊂ x , • - • 2 2r < α ⊂ Q x ,r,α (4.6)
and

q x ,r (x) := x , x - x 2 2r > α ≥ sup q x ,r (S) =: ϕ S,r (x ). (4.7)
Proof. Set p := proj S (x), d := d S (x) and u := d -1 (px). Applying Proposition 4.1 with ρ := 2r > d gives some β ∈ R such that for every y ∈ S,

u , x + d 2 2r < β < u , y + 1 2r p -y 2 . (4.8)
Now, observe that for all y ∈ X,

u , y + 1 2r y -p 2 = u - 1 r p, y + 1 2r ( y 2 + p 2 ). (4.9)
Putting the second inequality of (4.8) and the equality (4.9) together gives for all y ∈ S,

1 r p -u , y - y 2 2r < p 2 2r -β.
On the other hand, the first inequality of (4.8) and the equality (4.9) yield

1 r p -u , x - x 2 2r > p 2 2r -β.
Then, x := 1 r pu = Γ S,r (x) and α := p 2 2r -β fulfill the inclusion in (4.6) along with the estimates (4.7). The proof is complete.

Remark 4.2 It should be noted that we have a similar separation property as (4.6) with reversed inequalities. Indeed, consider any r-prox-regular subset S of X with r ∈]0, +∞] and x ∈ U r (S) \ S. It is straightforward to check that -S is also r-prox-regular and that the inclusion -x ∈ U r (-S) \ -S holds. Hence, according to Theorem 4.3, we can write for some u ∈ X and some real β

-S ⊂ u , • - • 2 2r < β and u , -x - x 2 2r > β.
Setting x := -u and α := -β, we obtain

S ⊂ x , • + • 2 2r > α and x , x + x 2 2r < α.
As a direct application, we can extend the classical fact that a lower semicontinuous convex function has continuous affine minorants.

Theorem 4.4 Let f : X → R ∪ {+∞} be a function with an r-prox-regular epigraph for some r ∈]0, +∞[. Then, there exist x ∈ X, α, θ ∈ R with θ < sup

X 1 r f such that x , x - x 2 2r + α ≤ f (x) f (x) 2r -θ for all x ∈ X.
Proof. We endow X × R with the canonical inner product and we still denote the associated norm by • . If f ≡ +∞ there is nothing to establish, so we may assume that f ≡ +∞. Fix any u ∈ X with f (u) < +∞ and choose any real 0 < ε < r. Note that (u, f (u)) ∈ epi f , (u, f (u) -ε) / ∈ epi f and Combining (4.10) and the inclusion (x, f (x)) ∈ epi f , we obtain

d epi f (u, f (u) -ε) ≤ ε < r.
u , x + θf (x) - 1 2r (x, f (x)) 2 ≤ α.
From this and (4.11), we get

u , x + θf (x) ≤ 1 2r (x, f (x)) 2 + α < 1 2r (x, f (x)) 2 + u , x + θc - 1 2r (x, c) 2 , hence θf (x) < θc + 1 2r (f (x) 2 -c 2 ).
On the other hand, using the inequality f (x) > c, we arrive to

θ < 1 2r (f (x) + c) < 1 r f (x) ≤ sup X 1 r f.
Finally, taking any x ∈ X with f (x) = +∞ (otherwise, the desired inequality is obvious), it remains to apply (4.10) with the inclusion (x, f (x)) ∈ epi f to obtain

u , x - 1 2r x 2 ≤ α + f (x)( 1 2r f (x) -θ).
The proof is complete.

Remark 4.3 Any r -1 -semiconvex continuous function f : X → R has its epigraph uniformly prox-regular with constant r. Indeed, since g := f + 1 2r • 2 is a continuous convex function, it is locally Lipschitz on int(dom f ) = X as well as the function h : X → R defined by

h(x, t) := g(x) - x 2 2r
t for all (x, t) ∈ X × R.

From the definition of h it can be checked in a straightforward way that {h ≤ 0} = epi f.

On the other hand, we see that

∂ C h(x, t) = (x - x r , -1) : x ∈ ∂ C g(x)
for all x ∈ X.

Since g is convex, its Clarke subdifferential ∂ C g is monotone, hence ∂ C h is 1 r -hypomonotone. It remains to see that for every (x , x) ∈ X 2 , (xx r , -1), (0, 1) = -1

to apply [1, Theorem 4.1] and get the r-prox-regularity of {h ≤ 0} as desired.

Theorem 4.3 makes clear the important role played by sublevel sets (Q x ,r,α ) α of the function q x ,r . Such sets are in general nothing but complement of suitable open balls. Indeed, fix any x ∈ X, α ∈ R and r ∈]0, +∞[. Let x ∈ X. Thanks to the elementary equality

xrx 2 = r 2 x 2 -2r x , x + x 2 it is easy to observe that

x ∈ Q x ,r,α ⇔ x -rx 2 ≥ r 2 x 2 -2rα.
Hence, we get the following equivalence

x ∈ Q x ,r,α ⇐⇒ x ∈ B(rx , r 2 x 2 -2rα) if r 2 x 2 -2rα > 0 x ∈ X otherwise.
If α = ϕ S,r (x ) for some closed subset S of X, the above equivalence along with the following equalities

r 2 x 2 -2rα = r 2 x 2 -2r sup x∈S ( x , x - x 2 2r ) = -sup x∈S (-r 2 x 2 + 2r x , x - x 2 2r ) = inf x∈S rx -x 2 = d 2 S (rx ) ≥ 0 ensure that Q x ,r,α = X \ B(rx , d S (rx )) if rx / ∈ S, X otherwise.
We state the above descriptions of Q x ,r,α in the following proposition.

Proposition 4.2 Let x ∈ X, α ∈ R and r ∈]0, +∞[. One has

Q x ,r,α = X \ B(rx , r 2 x 2 -2rα) if r 2 x 2 -2rα > 0, X otherwise.
In particular, if ϕ S,r (x ) = α ∈ R for some closed subset S of X, one has

r 2 x 2 -2rα = d 2 S (rx )
and

Q x ,r,α = X \ B(rx , d S (rx )) if rx / ∈ S, X otherwise.
Remark 4.4 Let S be an r-prox-regular subset of X with r ∈]0, +∞[. Fix x ∈ bdry X S and choose any x ∈ N (S; x). According to Theorem 2.1, for every x ∈ S, we have

x , x -x ≤ x 2r x -x 2 ,
which can be rewritten as

S ⊂ x ∈ X : x x -x ≤ x 2r x -x 2 .
If x = 0, the latter inclusion is reduced to the evident one S ⊂ X, so assume that x = 0. Set u := x x + x r and α := x x , x + x 2 2r and note that for each

x ∈ S x , x -x ≤ x 2r x -x 2 ⇔ x , x ≤ x , x + x 2r ( x 2 -2 x, x + x 2 ) ⇔ u , x - x 2 2r ≤ α, hence S ⊂ Q u ,r,α .
On the other hand, it is easy to check that

ρ := r 2 u 2 -2rα = r 2 > 0.
Then, by virtue of Proposition 4.2

S ⊂ X \ B(r x x + x, r).
Hence, the set X \ Q u ,r,α is nothing but an r-ball which realizes the proximal normal vector x

x to S at x.

Given x ∈ X, r ∈]0, +∞[ and α ∈ R such that ρ = r 2 x 2 -2rα > 0, from the above description of the closed set Q x ,r,α in Proposition 4.2 says that

Q x ,r,α = X \ B(rx , √ ρ), hence Q x ,r,α is √ ρ-prox-regular, d(x, Q x ,r,α ) = ( √ ρ -x -rx ) + for all x ∈ X S U r (S) x 1 rx 1 B 1 x 5 rx 5 B 5 rx 6 B 6 x 6 x 3 B 3 rx 3 x 2 B 2 rx 2 x 4 B 4 rx 4
∈ X \ Q x ,r,α Proj Q x ,r,α (x) = S(rx , √ ρ) if x = rx , √ ρ x-rx (x -rx ) + rx otherwise.
It is worth noticing that the above uniformly prox-regular sets Q x ,r,α are never weakly sequentially closed and never weakly ball-compact whenever the Hilbert space X is infinitedimensional. According to Proposition 4.2, we have rx -

x = √ ρ, hence with v := -w ∈ B ∇f (x), v = x -x, v = - √ ρ r = -δ.
A direct application of [START_REF] Adly | Preservation of Prox-Regularity of Sets with Applications to Constrained Optimization[END_REF]Theorem 4.1] gives the r -prox-regularity of S = Q x ,r,α with r := rδ = √ ρ.

Remark 4.6 There is no hope for a kind of Mazur's lemma for prox-regular sets. Besides the non-sequential weak closedness observed above for the uniformly prox-regular sets Q x ,r,α , let us mention that there exist uniformly prox-regular sets which are weakly sequentially closed but not weakly closed. Indeed, with (e n ) n∈N the natural hilbertian basis of l 2 R (N), the set S := { √ ne n : n ∈ N} is weakly sequentially closed (in particular, strongly closed), not weakly closed [7, Example 3.31] and uniformly prox-regular thanks to the equality √ pe p -√ qe q = √ p + q ≥ √ 3 for all p, q ∈ N with p = q.

Ball separation

This section is devoted to a ball separation for a prox-regular set and a strongly convex set in X.

Definition 5.1 Let C be a nonempty closed subset of X and s ∈]0, +∞[. One says that C is s-strongly convex (or strongly convex with constant s) provided that it is an intersection of closed balls with radius s > 0, otherwise stated there exists a nonempty set L such that

C = x∈L B[x, s].
Let us recall some basic features for such sets. (d) For all x, x ∈ C and for all v ∈ N (C; x), one has

v, x -x ≤ - v 2s x -x 2 .
Furthermore, any non-singleton strongly closed set has nonempty interior. We will need another property which can be found in G.E. Ivanov's book [23, Theorem 1.12.3]. Theorem 5.2 Let S 1 and S 2 be nonempty sets in X. If S 1 is r-prox-regular with r ∈ ]0, +∞] and S 2 is s-strongly convex with 0 < s < r. Then the set S 1 + S 2 is closed.

The result in the next proposition is known (see G.E. Ivanov [START_REF] Ivanov | Weakly Convex Sets and Functions: Theory and Applications[END_REF]Theorem 1.12. 4]). We provide a simple proof different from that in [START_REF] Ivanov | Weakly Convex Sets and Functions: Theory and Applications[END_REF]. Observe first that, given two sets S 1 , S 2 in X and x * ∈ N P (S 1 + S 2 ; x), for any x i ∈ S i with x = x 1 + x 2 it directly results from the definition of proximal normal that x * ∈ N P (S 1 ; x 1 ) ∩ N P (S 2 ; x 2 ). Proposition 5.1 Let S 1 be an r-prox-regular set in X and S 2 be a strongly convex set in X with constant s ∈]0, r[. Then, the set S 1 + S 2 is (rs)-prox-regular.

Proof. We know by Theorem 5.2 that S 1 + S 2 is closed. Fix any x ∈ S 1 + S 2 , so there are x i ∈ S i , with i = 1, 2, such that x = x 1 + x 2 . Take any x * ∈ N P (S 1 + S 2 ; x) and any y ∈ S 1 + S 2 . Choose y i ∈ S i , with i = 1, 2, such that y = y 1 + y 2 . We have x * ∈ N (S 2 ; x 2 ) ∩ N P (S 1 ; x 1 ). Using both (d) in Theorem 5. This combined with (5.1) yields x * , y 1 + y 2 -x ≤ x * 2(r-s) y 1 + y 2 -x 2 , which ensures the (rs)-prox-regularity of S 1 + S 2 by (b) in Theorem 2.1 again.

We can now give two ball separation results in the next two theorems between a proxregular set and a strongly convex set. Similar results were also established by J.-P. Vial Proof. By Theorem 5.1 we know that int C = ∅. Let us first show that (-x + int C) ∩ T B (S; x) = ∅. Indeed, suppose there is some h in the latter intersection. There are sequences (t n ) n tending to 0 with 0 < t n < 1 and (h n ) n converging to h in X such that x + t n h n ∈ S for all n ∈ N. For n large enough, say n ≥ N , we have h n ∈ -x + int C, hence (since 0 ∈ cl (-x + int C)) we see that t n h n ∈ -x + int C. It ensues that x + t N h N ∈ S ∩ int C, which is a contradiction.

By the emptiness of the above intersection and the Hahn-Banach separation property there is some v ∈ X with v = 1 such that < s + g + (ȳx) 1 -rs g , then y -(x + rv) < s + g + g r-s g -1 = r, which confirms that B(ȳ + sv, s + g) ⊂ B(x + rv, r). This finishes the proof of the theorem.

  r ∈]0, +∞] be an extended real. Through the distance function, we can define the (open) r-tube of S as the set Tube r (S) := U r (S) \ S where U r (S) is the (open) r-enlargement of S U r (S) := {x ∈ X : d S (x) < r} , the (closed) r-exterior of S as the set Exte r (S) := {x ∈ X : d S (x) ≥ r} and the set of points at exact r-distance to S D r (S) := {x ∈ X : d S (x) = r} . If r = +∞, such sets become U ∞ (S) := X and Exte ∞ (S) = ∅ = D ∞ (S).

Theorem 2 . 1

 21 Let S be a nonempty closed subset of X and let r ∈]0, +∞]. Consider the following assertions. (a) The set S is r-prox-regular. (b) For all x, x ∈ S, for all v ∈ N P (S; x), one has v, xx ≤ 1 2r v xx 2 .

2 2r,

 2 Setting y := x + x r = x r and ϕ S,r (y ) := sup u∈S y , uu we arrive to S ⊂ y , • -• 2 2r ≤ ϕ S,r (y ) and y , xx 2 2r ≥ ϕ S,r (y ).

  (a ) is a direct consequence of what precedes. (b) and (b ): The implication claimed by (b) is a direct consequence of (a) and of the characterization (2.1) of proximal normal vectors. Similarly, (b ) follows from both (a ) and (2.1). (c) and (c ): These assertions follows from (a) and (a ) respectively. Now, assume that the set S is r-prox-regular. (d) and (d ): It is enough to establish (d). If there is z ∈ S such that u ∈ N (S; z) ∩ B + z r , then Theorem 2.1 ensures that

  (k) It suffices to apply again the Fenchel-Moreau theorem. (l) It suffices to use the fact that for a proper convex lower semicontinuous function h : X → R, we have (see, e.g., [28, Proposition 10.1]) for every x, u ∈ X (x, u) ∈ gph ∂h ⇔ (u, x) ∈ gph ∂h . It remains to establish assertions (m) -(p). Note that (n) (resp. (p)) is a direct consequence of the Fenchel-Moreau theorem and the equality provided by (m) (resp. (o)).

  andx , x > α ≥ sup y∈Sx , y =: σ(x , S).

Lemma 4 . 2

 42 Let S be an r-prox-regular subset of X with r ∈]0, +∞[ and let x ∈ Tube r (S).The following hold with t := d S (x) r ∈]0, 1[. (a) One has x = trΓ S,r (x) + (1 -t)proj S (x) and Γ S,r (x) -x r = 1 -t, in particular x ∈]proj S (x), rΓ S,r (x)[. (b) One has the inclusions rΓ S,r (x) = (1 -t -1 )proj S (x) + t -1 x ∈ D r (S) and proj S (x) ∈ Proj S (rΓ S,r (x)). (c) One has Tube r (S) = u∈Tuber(S)]proj S (u), rΓ S,r (u)[. (d) For all y ∈]proj S (x), rΓ S,r (x)[, one has Γ S,r (x) = Γ S,r (y). (e) For every s ∈]0, r[, one has Λ r (S) := rΓ S,r Tube r (S) = (1 -r s )proj S (y) + r s y : y ∈ D s (S) . (f ) For every s ∈]0, r[, one has Tube r (S) = u∈Ds(S) ]proj S (u), rΓ S,r (u)[. Proof. (a) It can be checked in a straightforward way. (b) Observe from Remark 2.1 that with p := proj S (x) and d := d S (x) p ∈ Proj S p + r xp d = Proj S rΓ S,r (x) . Consequently, we have d S rΓ S,r (x) = rΓ S,r (x) -p = t -1 xp = r. (c) According to the assertion (a) above, we know that Tube r (S) ⊂ u∈Tuber(S) ]proj S (u), rΓ S,r (u)[. Let us show the converse inclusion. Fix any u ∈ Tube r (S) and any y ∈]proj S (u), rΓ S,r (u)[. Set d := d S (u) and p := proj S (u). Let λ ∈]0, 1[ such that y = λp + (1 -λ)rΓ S,r (u). We obviously have

  Consequently, we have Tube r (epi f ) ∩ (dom f × R) = ∅. Fix any (x, c) ∈ Tube r (epi f ) with f (x) < +∞. According to Theorem 4.3, there are (u , θ) ∈ X × R and α ∈ R such that epi f ⊂ {ϕ ≤ α} and (x, c) ∈ {ϕ > α}with ϕ : X × R → R defined by ϕ(x, t) := u , x + θt -1 2r (x, t) 2 for all (x, t) ∈ X × R.This says that u , x + θt -1 2r (x, t) 2 ≤ α for all (x, t) ∈ epi f (

Figure 1 :

 1 Figure 1: An r-prox-regular set with its r-open enlargement and some r-balls associated to proximal normals. For the point x 2 , the complement of the ball B 2 illustrates the separation property provided by Theorem 4.3.

Remark 4. 5 2 2r

 52 The √ ρ-prox-regularity of Q x ,r,α can also be seen through [1, Theorem 4.1]. Indeed, keep the assumption ρ > 0 and define the function f : X → R by setting f (u) := x , uu α for all u ∈ X, so S = {f ≤ 0}. Set δ := √ ρ r > 0 and consider any x ∈ bdry Q x ,r,α . Choose w ∈ B such that sup v∈B rxx, v = rxx, w = rxx .

Theorem 5 . 1

 51 Let C be a nonempty closed bounded convex subset of X with C = X and let s ∈]0, +∞[. The following assertions are equivalent: (a) The set C is s-strongly convex. (b) For all (x, v) ∈ X 2 with x ∈ bdry C and v ∈ N (C; x) ∩ S, one has C ⊂ B[xsv, s]. (c) For all x ∈ bdry C and for all (x, v) ∈ X 2 with x ∈ bdry C and v ∈ N (C; x) ∩ S, one has v, xx ≤ -1 2s xx 2 .

  The equivalence (a) ⇔ (b) in the above theorem was proved by J.-P. Vial[START_REF] Vial | Srong convexity of sets and functions[END_REF] Theorem 1] and by E.S. Polovinkin and M.V. Balashov[START_REF] Polovinkin | Elements of Convex and Strongly Convex Analysis[END_REF] Theorem 4.1.2], and (c) is a translation of (b).

1 ) 2 ≤

 12 1 and (b) in Theorem 2.1 we obtain x * , y 1 + y 2 -x ≤ x Set a := y 1 -x 1 and b := y 2 -x 2 and note that a 2 + b 2 + 2 a, b = a + b 2 .

[ 37 ,

 37 Theorem 5.1] in finite-dimensional Euclidean spaces and by G.E. Ivanov [23, Theorem 1.18.2] in Hilbert spaces. Theorem 5.3 Let S be an r-prox-regular set of the Hilbert space X with r > 0 and C be a non-singleton closed set in X which is r-strongly convex with C ∩S = {x} and x ∈ bdry C. Then one has the ball separation property C ⊂ B[xrv, r] and B(xrv, r) ∩ S = ∅.

Theorem 5 . 4

 54 v, -x + z < 0 ∀z ∈ int C and v, h ≥ 0 ∀h ∈ T B (S; x) = T C (S; x), keep in mind that T C (S; x) is a closed convex cone and 0 ∈ cl (-x+C). Using the equalities T C (S; x) • = N C (S; x) = N P (S; x), it results that v ∈ N (C; x) and -v ∈ N P (S; x), thus by (b) in Theorem 5.1 and by Definition 2.1 combined with (2.4) we haveC ⊂ B[xrv, r] and S ∩ B(xrv, r) = ∅,which is the desired separation property. Let S be an r-prox-regular set of the Hilbert space X with r > 0 and C be a closed set in X which is s-strongly convex with 0 < s < r. For g := gap(S, C), where gap(S, C) := inf{ xy : x ∈ S, y ∈ C}, assume that 0 < g < rs. Then there exists a ∈ X such that C ⊂ B[a, s] and S ∩ B(a, s + g) = ∅, and the latter equality entails in particular S ∩ B[a, s] = ∅. Proof. By Proposition 5.1 the set S -C is (rs)-prox-regular and 0 ∈ S -C with d(0, S -C) = g < rs. The point 0 then admits a (unique) nearest point z ∈ S -C. Write z = xȳ with x ∈ S and ȳ ∈ C, so g = xȳ . Note that x (resp. ȳ) is a nearest point of ȳ (resp. x) in S (resp. C). Setting v := (ȳx)/ ȳx , we see that v belongs to N (C; ȳ) and N P (S; x). By (b) in Theorem 5.1 and by Definition 2.1 combined with (2.4) we have C ⊂ B[ȳ + sv, s] and S ∩ B(x + rv, r) = ∅. It remains to prove the inclusion B(ȳ+sv, s+g) ⊂ B(x+rv, r). Take any y ∈ B(ȳ+sv, s+g) and note that y -(x + rv) = y -(ȳ + sv) + (ȳx + svrv)

Acknowledgement. The second author has received funding from the European Union's Horizon 2020 Research and Innovation Programme under the Marie Sklodowska-Curie Grant Agreement No 823731 CONMECH.


Sylvain Rodat
email: sylvain.rodat@promes.cnrs.fr.

Stéphane Abanades

Jean-Louis Sans

Gilles Flamant

A pilot-scale solar reactor for the production of hydrogen and carbon black from methane splitting

A pilot-scale solar reactor was designed and operated at the 1 MW solar furnace of CNRS for H2 and carbon black production from methane splitting. This constitutes the final objective of the SOLHYCARB EC project. The reaction of CH4 dissociation produces H2 and carbon nanoparticles without CO2 emissions and with a solar upgrade of 8% of the high heating value of the products. The reactor was composed of 7 tubular reaction zones and of a graphite cavity-type solar receiver behaving as a black-body cavity. Temperature measurements around the cavity showed a homogeneous temperature distribution. The influence of temperature (1608K-1928K) and residence time (37-71 ms) on methane conversion, hydrogen yield, and carbon yield was especially stressed. For 900 g/h of CH4 injected (50% molar, the rest being argon) at 1800K, this reactor produced 200 g/h H2 (88% H2 yield), 330 g/h CB (49% C yield) and 340 g/h C2H2 with a thermal efficiency of 15%. C2H2 was the most important by-product and its amount decreased by increasing the residence time. A 2D thermal model of the reactor was developed. It showed that the design of the reactor front face

could be drastically improved to lower thermal losses. The optimised design could reach 77% of the ideal black-body absorption efficiency (86% at 1800K), i.e. 66%. Keywords: Hydrogen production; Solar thermal energy; Methane dissociation; Solar reactor; Thermal simulation.

1.Introduction

Solar methane dissociation offers the possibility for the clean co-production of hydrogen and carbon black. The benefits of such a process can be found elsewhere [START_REF] Spath | Using a concentrating solar reactor to produce hydrogen and carbon black via thermal decomposition of natural gas: feasibility and economics[END_REF][START_REF] Pregger | Prospects of solar thermal hydrogen production processes[END_REF]. It appears as an alternative to the steam methane reforming and the furnace process [START_REF] Lockwood | Parametric Study of a Carbon Black Oil Furnace[END_REF] dedicated to the conventional production of hydrogen and carbon black, respectively. The solar process avoids both CO2 emissions from fossil fuel combustion required to carry out the endothermic reaction and from the reaction of steam reforming thus avoiding 13.9 kg-equivalent CO2/kg H2 produced as compared to conventional processes [START_REF] Dahl | Rapid solar-thermal dissociation of natural gas in an aerosol flow reactor[END_REF]. Indeed, the energy supplied by fossil fuel combustion is replaced by concentrated solar energy and methane cracking results in solid carbon and hydrogen only. The theoretical energy balance of the reaction is reported in Figure 1. It shows that the High Heating Value (HHV) of methane is upgraded by solar energy in an amount equal to the enthalpy change of the reaction (76 kJ/mol). Finally, solar energy represents 8% of the HHV of the products, hydrogen 59% and carbon 41%. Anyway, it must be pointed out that the HHV of carbon is lost since it is not destined for energy production but rather for direct industrial application such as tyres or batteries production [START_REF] Donnet | Carbon Black, Second edition, Revised and expanded[END_REF].

Solid carbon could also be an opportunity for a safe carbon storage with a possible future use if ecological situation is favourable [START_REF] Muradov | Green'' path from fossil-based to hydrogen economy: An overview of carbon-neutral technologies[END_REF]. The sensible heat of the products can also be recovered.

Solar chemical reactors devoted to the solar thermal dissociation of methane were proposed at small scale. Maag et al. [START_REF] Maag | Solar thermal cracking of methane in a particle-flow reactor for the co-production of hydrogen and carbon[END_REF] developed a direct heating solar reactor seeded with particles. A numerical model was also proposed accounting for the unsteady mass and energy conservation equations, coupling convective heat and mass transfers, radiative heat transfer, and chemical kinetics for a two-phase solid-gas flow [START_REF] Maag | Particle-gas reacting flow under concentrated solar irradiation[END_REF]. Fairly good agreement between the model and the experimental results were obtained. Direct solar heating was also experienced by Kogan et al. [START_REF] Kogan | Production of hydrogen and carbon by solar thermal methane splitting. I. The unseeded reactor[END_REF][START_REF] Kogan | Production of hydrogen and carbon by solar thermal methane splitting. III. Fluidization, entrainment and seeding powder particles into a volumetric solar receiver[END_REF] with a tornado flow configuration and Hirsch et al. [START_REF] Hirsch | Solar hydrogen production by thermal decomposition of natural gas using a vortex-flow reactor[END_REF]. Indirect heating reactors based on tubular designs were proposed by Dahl et al. [START_REF] Dahl | Solarthermal dissociation of methane in a fluid-wall aerosol flow reactor[END_REF] and Rodat et al. [START_REF] Rodat | High-Temperature Solar Methane Dissociation in a Multitubular Cavity-Type Reactor in the Temperature Range 1823-2073 K[END_REF]. Direct heating solar reactors provide better heat transfer to the reaction zone but the window protection against particle deposition is a challenging task. It seems from the current state-of-the-art that indirect heating solar reactors are more mature for scaling up. If the reactor was operated with very high gas velocities (0.3 to 0.8 Mach [START_REF] Donnet | Carbon Black, Second edition, Revised and expanded[END_REF]) and with longer tubes (to keep the same gas residence time), the carbon deposition could be alleviated. The investigation of reaction kinetics was also addressed [START_REF] Dahl | Intrinsic kinetics for rapid decomposition of methane in an aerosol flow reactor[END_REF][START_REF] Rodat | Kinetic modelling of methane decomposition in a tubular solar reactor[END_REF][START_REF] Younessi-Sinaki | Kinetic model of homogeneous thermal decomposition of methane and ethane[END_REF]. Reviews on solar methane cracking [START_REF] Ozalp | Solar decomposition of fossil fuels as an option for sustainability[END_REF] and catalytic methane decomposition using metal and carbonaceous materials [START_REF] Abbas | Hydrogen production by methane decomposition: A review[END_REF] were recently published. Nevertheless, the previous developed reactors do not exceed 10 kW scale. In order to acquire more experience toward a potential industrial application, a 50 kW multi-tubular solar reactor was constructed, tested and simulated in the framework of the European project SOLHYCARB. This 50 kW scale receiver is a step forward in a future industrial application of the process in a solar chemical plant based on a reactor set at the top of a central solar tower receiver and equipped with a secondary concentrator. In the various numerical models previously proposed, special attention was paid to detail the reaction zone.

For example, Caliot et al. [START_REF] Caliot | Effects of non-gray thermal radiation on the heating of a methane laminar flow at high temperature[END_REF] proposed a model describing the absorption of IR radiations by methane, Kogan et al. [START_REF] Kogan | Production of hydrogen and carbon by solar thermal methane splitting. IV. Preliminary simulation of a confined tornado flow configuration by computational fluid dynamics[END_REF] published a detailed study on the numerical simulation of their tornado flow configuration, but only few works addressed the global design of an optimised pilot-scale receiver [START_REF] Maag | Solar thermal cracking of methane in a particle-flow reactor for the co-production of hydrogen and carbon[END_REF][START_REF] Ozalp | CFD analysis on the influence of helical carving in a vortex flow solar reactor[END_REF], especially in terms of chemical and thermal reactor efficiencies. This paper presents the experimental chemical performances and energy efficiencies of the pilot-scale solar reactor, and it also focusses on the possible improvements of the receiver predicted by CFD simulations.

2.Experimental set-up

The reactor was designed for a nominal power of 50 kW of incident solar power (Figure 2). The reactor body is made of an aluminium shell (800x780x505 mm) and a water-cooled front face with a 13 cm-diameter aperture to let concentrated solar radiation entering within the reactor cavity. The radiations are absorbed by the graphite cavity (360x400x300 mm) that approaches black-body behaviour. To avoid contact of graphite with the oxidizing atmosphere, the aperture is protected by a domed quartz window (outer diameter of 360 mm) swept by a nitrogen flow. The hemispherical shaped window is advantageous in comparison to a plane window because it is not close to the hot cavity and the focal zone, which avoids window overheating and subsequent irreversible damage. The space between the graphite cavity and the aluminium shell is filled with three different insulating layers to limit conduction losses. It is composed of a 85 mm-thick graphite felt in contact with the cavity (=0.53 W.m -1 .K -1 at 1873K), a 50 mm-thick intermediate refractory ceramic fiber operating up to 1873K (62% Al2O3, 30% SiO2, =0.35 W.m -1 .K -1 at 1673K), and a 50 mm-thick outer microporous insulator operating up to 1273K (20% ZrO2, 77.5% SiO2, 2.5% CaO, =0.044 W.m -1 .K -1 at 1073K). Seven graphite tubes (800 mm length, 26 mm OD, 18 mm ID) cross the graphite cavity horizontally and they are heated both by direct solar radiation and by IRradiation from the hot graphite cavity walls. In order to maintain the position of the graphite cavity and to avoid mechanical stress on the tubes, four alumina tubes, positioned vertically as supports at the bottom, hold the graphite cavity.

The experimental equipment is schemed in Figure 3. Each tube is fed with a mixture of argon and methane thanks to two mass-flow controllers dedicated to each tube (total of 14 massflow controllers, Brooks Instruments model 5850 S, range 0-10 NL/min for methane and 0-20 NL/min for argon). Each tube entrance is equipped with an absolute pressure sensor. At the exit, the products (particles and gases) from the tubes are collected and cooled down. Then, they are directed toward a filter composed of 6 bags (diameter: 127 mm, length: 1000 mm) enabling the separation of the carbon black from the gaseous phase that is evacuated to the vent. Before this filter, a pump samples a part of the products toward a secondary filter and then to the gas analysis system. This sampling system provides a fast response time of the analysis, since there is no need to wait for the sweeping of the whole volume of the primary filter (about 0.8 m 3 ) to get stable measurements. The gas analysis system is composed of an online analyser for measuring hydrogen and methane concentrations (NGA 2000 MLT3, resolution of 1% of the full scale, range: 0-70% for H2, 0-10% for CH4) as well as a gas chromatograph (micro GC Varian CP 4900) for identifying and quantifying the gas species in the course of the experiment. The micro GC is equipped with 2 channels: Channel 1 (MolSieve 5A PLOT 10M Backflush) for H2, N2, O2, CO, CH4; Channel 2 (PoraPLOT U 10M Backflush) for light hydrocarbons such as CH4, C2H6, C2H4, C2H2, and H2. These equipments were previously used during the testing of a 10 kW solar reactor prototype [START_REF] Rodat | Hydrogen production from solar thermal dissociation of natural gas: development of a 10 kW solar chemical reactor prototype[END_REF].

The control of the temperatures in the reactor is provided by five thermocouples and one solar blind optical pyrometer (wavelength: 5.14 µm) (Figure 4). Two type B thermocouples are set in contact with the graphite cavity, one at the top (B_top), another at the back (B_back). At about the same location, K-thermocouples (K_top and K-back) measure the temperature at 5 cm depth (from the aluminium shell surface) in the insulated zone. In addition, a type B thermocouple is inserted 24 cm inside the lowest graphite tube on which the optical pyrometer is also pointing, thus giving a redundant temperature measurement of the reaction zone by two different means. A K-thermocouple and a pressure sensor are also positioned on the outlet gas path after the collector.

An experimental run is composed of two steps. The first step is the heating of the reactor under concentrated solar irradiation coming from the 1 MW solar furnace of CNRS-PROMES (the incoming power is regulated thanks to a shutter and to the number of sun-tracking heliostats, 1 MW corresponds to the maximum power of the solar facility with the shutter fully opened and 63 heliostats involved) [START_REF] Rodat | Hydrogen production from solar thermal dissociation of natural gas: development of a 10 kW solar chemical reactor prototype[END_REF]. During this period, the tubes are fed with pure argon till the targeted temperature is reached. Then, a mixture of argon and methane is injected once the temperature is stabilized and the operating conditions are maintained during about an hour to produce significant amounts of carbon black for further analysis and to be able to analyse the product characteristics with respect to fixed experimental conditions. The cleaning of the filter is realised with air back pulses and carbon is collected at the bottom of the filter at the end of the experiment. The whole experimental installation in operation is shown in Figure 5.

3.Experimental conditions

The experimental conditions are listed in Table 1. Methane mole fractions in the feed range from 25 to 50% in order to maintain sufficient dilution. The main results are recapped in Table 2. After each experimental run at given operating conditions (temperature, pressure and gas flow-rates fixed), the carbon was recovered in the filter so that carbon black samples were representative of specific conditions. About 100 g of sample was recovered after each run and was available for analysis and characterization. The specific work devoted to the analysis of the carbon black properties is not presented in the following. After a heating period of about 40 min under an argon flow in the tubes, the temperature of the reactor reaches 1700K. All the temperature sensors tend to indicate the same mean temperature in the range 1670K-1720K. The highest temperature is given by the pyrometer that points directly on the outer wall of a tube. At about the end of the experiments, the highest temperature is given by the thermocouple called B_back but its value is still very close to the one given by the pyrometer. It can be stated that the temperature is homogeneous around the cavity during the methane splitting period. The lowest temperature is given by the thermocouple inserted in a tube (B_tube) because it is cooled by the fresh reactants and by the endothermic reaction. During the heating period it can be noted that the temperature measurements are in the following order: Pyrometer > B_tube > B_back > B_top and the opposite ranking is observed during the passive cooling period. Small temperature variations can be observed at the beginning of the injection period, which can be correlated with a decrease of the DNI due to a cloud shadow on the heliostat field. However, it was possible to maintain the temperature around the targeted value by increasing the incoming power (increased shutter aperture). Concerning the K-thermocouples, they never reach a stable value during the injection period, which means that the insulation layers does not reach thermal steady state. The temperature even increases after the end of the reactor heating due to low thermal conduction and thermal inertia.

4.Results and discussion

4.1.Experimental results

When the targeted temperature is reached and stabilized, 31.5 NL/min argon and 10.5 NL/min methane are injected. H2 is rapidly detected at the exit along with residual CH4 (not dissociated). The temperature variations impact directly H2 and CH4 mole fractions, i.e. a temperature increase results in the H2 mole fraction increase and vice versa. Since the higher the temperature, the better the conversion, the trend is reverse for the CH4 mole fraction. Indeed, it decreases when the temperature increases. No continuous analysis of C2H2 in the off gas was available but the C2H2 mole fraction was rather constant: the chromatography analysis gave a 0.043 mean mole fraction. This value tended to decrease at the end of the injection period, which must be related to the H2 mole fraction increase and the CH4 mole fraction decrease. After 2400 s of experiment at constant temperature, the H2 mole fraction in the off-gas increases slightly as a result of tubes clogging. Carbon deposit in the tubes causes a pressure increase and thereby a residence time increase that favours better CH4 and C2H2

dissociation. After about one hour of methane injection, the methane flow-meters are successively stopped because of the progressive tube blocking.

The reactor performances are given in terms of methane conversion, hydrogen yield, and carbon yield [START_REF] Rodat | High-Temperature Solar Methane Dissociation in a Multitubular Cavity-Type Reactor in the Temperature Range 1823-2073 K[END_REF]. These parameters pertinently express the performances of the process giving the quantity of methane that has been transformed in the form of hydrogen and carbon, the two targeted products.

 The CH4 conversion gives the proportion of methane that has been dissociated and it is defined as:

4 4 4 CH 0, CH CH 0, CH F F.y F X   (1)
 The H2 yield is the proportion of methane that has been converted into hydrogen and it is calculated from:

4 2 2 CH 0, H H 2.F F.y Y  (2)
 The C yield is the proportion of methane that has been converted into solid carbon and it is expressed as:

4 6 2 4 2 2 2 4 4 CH 0, H C H C H C CH CH 0, C F ) F.y . 2 F.y . 2 F.y . 2 -(F.y F Y     (3)
where CH4 0,

F

is the inlet molar flow-rate of CH4, yi is the mole fraction of species i, and F is the total outlet gas flow-rate (including argon as buffer gas) obtained from: . ) y y y y (y -1

F F 6 2 4 2 2 2 2 4 H C H C H C H CH Ar      (4)
FAr is the molar flow-rate of Ar. In equations ( 3) and ( 4), polycyclic aromatic hydrocarbons (PAH) are neglected.

Figure 7 reports the results concerning the first experimental series (Ar: 31.5 NL/min, CH4:

10.5 NL/min) in terms of methane conversion, hydrogen yield, and carbon yield with increasing temperatures (1608K-1928K). The higher the temperature, the better the chemical performance criteria. Moreover, the CH4 conversion is always higher than the H2 yield and the C yield. This is because the methane dissociation is not complete: a part of the methane is not converted and another part is converted into other C2 hydrocarbons (mainly C2H2). Since the conversion of CH4 into C2H2 leads to the production of 1.5 mole of H2 per mole of CH4 without carbon production, the H2 yield is higher than the C yield. The intermediate C2H2 mainly affects the carbon yield. For temperatures above 1778K, complete methane conversion is achieved. It can also be noticed that similar runs lead to close results (runs 3 and 4).

4.2.Kinetic analysis

The kinetic parameters of the global reaction of CH4 dissociation were estimated on the basis of an isothermal plug-flow reactor model that is described by [START_REF] Trommer | Kinetic investigation of the thermal decomposition of CH4 by direct irradiation of a vortex-flow laden with carbon particles[END_REF]. A first order kinetic expression following an Arrhenius law was assumed, as proposed by Trommer et al. [START_REF] Trommer | Kinetic investigation of the thermal decomposition of CH4 by direct irradiation of a vortex-flow laden with carbon particles[END_REF]. The first series was considered since it shows the widest temperature range for a given gas flowrate. An activation energy of 196 (+/-17) kJ/mol and a pre-exponential factor in the range 1.57x10 7 s -1 -1.61x10 8 s -1 (accounting for the fit uncertainty) were identified. These values are in agreement with the ones obtained in a previous work on a 10 kW solar reactor [START_REF] Rodat | High-Temperature Solar Methane Dissociation in a Multitubular Cavity-Type Reactor in the Temperature Range 1823-2073 K[END_REF][START_REF] Rodat | Hydrogen production from solar thermal dissociation of natural gas: development of a 10 kW solar chemical reactor prototype[END_REF].

An activation energy in the range 172-205 kJ/mol and a frequency factor between 1.42x10 7 s -1 and 1.47x10 8 s -1 were obtained. This result highlights a weak dependency of the kinetics on the two different reactor designs (10 and 50 kW solar reactors). It is also consistent with the reported activation energies for heterogeneous (catalytic) methane decomposition reaction, which vary between 147 kJ/mol for activated carbons as particle loading [START_REF] Trommer | Kinetic investigation of the thermal decomposition of CH4 by direct irradiation of a vortex-flow laden with carbon particles[END_REF] and 236 kJ/mol for carbon black based catalysts [START_REF] Muradov | Catalytic activity of carbons for methane decomposition reaction[END_REF].

4.3.Energy efficiencies

The global reaction of direct methane decomposition is not able to predict the formation of C2H2 intermediate and a 2-step reaction mechanism was rather considered (Eq. ( 5) and ( 6)) to quantify realistic chemical energies associated with the formation of the main products (C, H2, and C2H2). In order to assess the performances of the reactor with respect to energy conversion efficiency, calorimetric measurements were performed at the cavity entrance (see Figure 4). Thus, the power reaching the back of the water-cooled front face was measured, which allowed the evaluation of the reactor thermochemical and thermal efficiencies, for the following two-step reaction mechanism:

CH4  3/2 H2 + ½ C2H2 endothermal, H1° = 188 kJ.mol -1 (5) C2H2  H2 + 2 C exothermal, H2° = -227 kJ.mol -1 (6)
The corresponding thermochemical and thermal efficiencies can be expressed as follows:

         (8)
Where Y is the fractional conversion of reaction ( 6) defined as Y=FH2/F0,C2H2 (F0,C2H2 is the molar flow-rate of acetylene issued from reaction ( 5)). X is the fractional conversion of reaction [START_REF] Donnet | Carbon Black, Second edition, Revised and expanded[END_REF]. Psolar is the solar power input at the cavity entrance (W), Cpi is the specific heat of species i (J.mol -1 .K -1 , function of temperature).

The incoming power varied between 23 and 37 kW (corresponding to mean flux densities between 1.7 and 2.8 MW/m²) depending on the reactor temperature (Figure 9). The resulting efficiencies are reported in Figure 10 for all the experimental runs. The two series can easily be identified. The first one shows lower efficiencies than the second one. This is due to the higher feed gas flow-rates for the second series, which led to higher thermal power conversion. In every cases, the thermochemical efficiency is smaller than the thermal efficiency because the latter accounts for the heating of argon and non converted methane.

The highest thermochemical and thermal efficiencies reported (13.5% and 15.2%, respectively) are achieved for the run #7, with 21 NL/min CH4 and Ar. As a result of a 50% limited dilution, the discrepancy between thermochemical and thermal efficiencies is lower than the one observed for the other runs that correspond to higher argon dilution (70 to 75%).

On the basis of run #7, the production at pilot-scale is 200 g/h H2 (88% H2 yield), 330 g/h CB (49% C yield), and 340 g/h C2H2. The production of C2H2 is significant and it could be either recovered as a valuable by-product or further dissociated provided that the residence time is increased.

5.Thermal simulation of the solar cavity receiver

A numerical model was proposed in order to identify and classify the main thermal losses associated with the design of the reactor. A global 2D axisymmetric model of the reactor was created with the commercial software Fluent (12.0.16). In order to get fast convergence and flexibility of the model, the reaction was simulated by a heat sink in the volume of the graphite cavity, which simplifies the tubular zones while taking into account the endothermic reaction. The mesh was created with Gambit (2.4.6) to simulate the whole receiver and to optimise its design. First, the reactor was simulated for comparison with a real experimental run. Then, a new design was proposed for a potential industrial up-scaled application.

The mesh (Figure 11) shows the 2D axisymmetric geometry that approaches the real 3D configuration. The thickness of the insulating layers is respected. However, the reactor is simulated as a cylindrical receiver. Moreover, although the model does not account for the tubes inside the cavity, the related thermal conduction losses are simulated by an equivalent quantity of matter around the outlet pipe that is cooled at the exit. The hemispherical window is not included, the diffuse solar irradiation is directly emitted by a transparent window at the reactor aperture (or at the reactor entrance depending on the case). The Discrete Ordinate model is used to solve the radiative transfer equation. The graphite cavity approaches an ideal blackbody, consequently the emissivity and diffuse fraction are set to 1. Perfectly diffused walls are thus assumed. The N2 sweeping gas flow-rate is kept constant (24 NL/min). A sensitivity study on the mesh was carried out in order to check independence of the results on the mesh size. A mesh composed of 21000 cells was chosen (element: quad, type: pave) with a smoother definition where the discrete ordinate model is solved to take into account the radiative heat transfer precisely. The convective heat transfer coefficient around the aluminium shell is set to 10 W/m²K. The top left corner of the cavity serves as the reference point for the cavity temperature.

First, the reactor was simulated for a given experimental condition (run #7: Ar and CH4 flowrates of 21 NL/min at 1798K). For this experiment, the thermal power required to heat the feed and carry out the two-step reaction was 4316 W. Calorimetric measurements showed that the power reaching the cavity entrance (see Figure 11) was 28.4 kW. Thus the reactor was modelled with this incident solar power. The influence of the heat sink on the cavity temperature was studied (Figure 12). Obviously, increasing the heat sink results in the cavity temperature decrease. For a heat sink of 4316 W, the cavity temperature is still 2020K. Calorimetric measurements were carried out at the cavity entrance for reasons of experimental implementation. The power at the aperture was different due the water-cooled front-face thickness. The model was thus used to estimate the power required at the aperture. To reach the same cavity temperature of 2020K, a power of 53 kW was required at the aperture (as compared to the 28.4 kW required at the cavity entrance, see Figure 11), which points out the high thermal losses due to the thickness of the water-cooled front face (35 mm). The corresponding flux density is 4.14 MW/m². Anyway, this predicted cavity temperature is about 200K higher than the experimentally measured value (about 1800K). This can be explained by additional thermal losses in the real configuration through the sighting portholes devoted to pyrometry measurements and other thermal bridges such as the alumina supports of the graphite cavity. It was shown previously that the reactor insulating layers are not at steady state whereas the simulation is steady, so the transient heat demand also contributes to extra thermal losses. In addition, the approximation of the geometry assumed cylindrical, which especially leads to a smaller water-cooled front face, may explain the discrepancy that corresponds to an additional heat sink of 6 kW (Figure 12).

In the following, these additional thermal losses were neglected since they can be avoided.

The thermal sink was thus considered to be directly connected to the thermal power required to heat the feed and carry out the two-step reaction mechanism. Consequently, the possible maximum heat sink was 10.3 kW (Figure 12) for maintaining the temperature at about 1800K (mean temperature measured experimentally) with 53 kW entering at the aperture. Thus, the thermal efficiency of the reactor was determined to be 19% if the whole heat sink is devoted to the reactants heating and the reaction enthalpy.

The main heat losses were identified to be at the water-cooled front face (66%) and at the aperture (9%). To diminish the heat losses at the aperture, higher solar concentration ratio should be used, but this requires a highly efficient solar concentrating system (capable to provide concentrated solar flux over 4.14 MW/m²). Since the most important thermal losses concern the front face, an optimisation of this component is proposed. It could be better insulated assuming the design represented in Figure 13. The boundaries are the same as for the previous design, but the front face is better insulated and includes a secondary concentrator shape (nevertheless, the aperture diameter is kept the same as in Figure 11). The compound parabolic concentrator (CPC) requires a specifically designed concentrating system as a function of its aperture angle. The CPC is water-cooled and its temperature is fixed at 300K. For 53 kW entering power at the aperture, a temperature of 1800K is obtained for a heat sink of 35 kW, resulting in a thermal efficiency of 66%. In addition, a homogeneous temperature of graphite is obtained. This specific design of the front face permits to increase drastically the thermal efficiency of the reactor. For comparison, the absorption efficiency of a perfectly insulated black-body cavity at 1800K irradiated through the same aperture with 53 kW is 86% according to the following expression [START_REF] Steinfeld | Encyclopedia of physical science & technologie[END_REF]:

C DNI T T Q Q Cavity actor Radiation Absorption . ) ( 1 1 4 0 4 Re        (9)
with σ, the Stephan Boltzmann constant (5.67.10 -8 W.m -2 .K -4 ), Tcavity, the cavity temperature and T0=298.15K.

Accordingly, the optimised design reaches 77% of the ideal black-body cavity absorption efficiency. If all the power can be transferred to the reactants, this could result in the production of 1.6 kg/h H2, 2.7 kg/h CB and 2.8 kg/h C2H2 (from Eq. 8 with a thermal efficiency of 66%) assuming similar chemical performances as the ones obtained with 21 NL/min CH4 and 21 NL/min Ar at 1798K (run #7). Such theoretical production corresponds to about 8 times the experimental production.

6.Conclusion

A pilot-scale solar reactor was tested to turn methane into hydrogen and carbon. For the investigated experimental conditions, CH4 conversions between 72% and 100% and H2 yields in the range 57%-88% were reached. Anyway, the carbon yield never exceeded 63%.

Thermochemical and thermal efficiencies up to 13.5% and 15.2% were achieved, respectively. In spite of a relatively low carbon yield due to the production of significant amounts of C2H2, representative quantities of carbon black were recovered for further characterisation. The carbon black samples must be analysed in order to bring into light correlations between experimental conditions and carbon black properties.

Thermal simulation of the cavity receiver was proposed to evaluate its thermal behaviour.

Results showed that 66% of the incoming power was lost through the water-cooled front face as a result of the high temperature gradient between the isothermal cavity at about 1800K and the front face at 300K. In order to address this issue, an optimized design of the front face was proposed and simulated: it reaches 77% of the ideal black-body absorption efficiency but it requires a specifically designed solar concentrating system.

Figure 6

 6 Figure 6 displays typical temperature measurements, H2 and CH4 off-gas mole fractions, DNI

Figure 8

 8 Figure 8 reports the results related to the second experimental series (Ar: 49 NL/min, CH4: 21

F

  Figure captions:

Figure 1 :Figure 2 :Figure 3 :Figure 4 :Figure 5 :Figure 6 :Figure 7 :Figure 8 :Figure 9 :Figure 10 :Figure 11 :Figure 12 :Figure 13 :

 12345678910111213 Figure 1: Energy balance of the solar methane cracking

Acknowledgements

This study was funded by the European Project Solhycarb (2006-2010, Contract SES-CT2006-19770). The authors wish to thank Olivier Prévost and Marc Garrabos for their technical support during the solar reactor manufacturing and operation.

Nomenclature:

Cpi Specific heat of species i (J/mol.K, function of temperature) DNI Direct normal irradiation (W/m²)