Athanase Papadopoulos

Marc Troyanov

Weak Minkowski Spaces

Keywords: AMS classification: 51B20, 53C70, 53C22, 53C60 Weak Minkowski space, Minkowski geometry, norm, Hilbert geometry, Funk geometry, weak norm, Mazur-Ulam theorem, Desarguesian space, Busemann G-space

We define the notion of weak Minkowski metric and prove some basic properties of such metrics. We also highlight some of the important analogies between Minkowski geometry and the Funk and Hilbert geometries.

Introduction

In the last decade of the 19 th century, Hermann Minkowski initiated new geometric methods in number theory, which culminated with the celebrated Geometrie der Zahlen published in 1910 [START_REF] Minkowski | Geometrie der Zahlen[END_REF]. Minkowski's work is referred to several times by David Hilbert in his 1900 ICM lecture [START_REF] Hilbert | The English translation was also reprinted in "Mathematical developments arising from Hilbert problems[END_REF], in particular in the introduction, where he declares:

The agreement between geometrical and arithmetical thought is shown also in that we do not habitually follow the chain of reasoning back to the axioms in arithmetical, any more than in geometrical discussions. On the contrary we apply, especially in first attacking a problem, a rapid, unconscious, not absolutely sure combination, trusting to a certain arithmetical feeling for the behavior of the arithmetical symbols, which we could dispense with as little in arithmetic as with the geometrical imagination in geometry. As an example of an arithmetical theory operating rigorously with geometrical ideas and signs, I may mention Minkowski's work, Die Geometrie der Zahlen.

Regarding the influence of this book on the birth of metric geometry, let us mention the following, from the paper [START_REF] Busemann | Novel results in the geometry of geodesics[END_REF] by Busemann and Phadke, p. 181:

Busemann had read the beginning of Minkowski's Geometrie der Zahlen in 1926 which convinced him of the importance of non-Riemannian metrics.

An early result of Minkowski in that theory, related to number theory, states that any convex domain in R 2 which is symmetric around the origin and has area greater than four contains at least one non-zero point with integer coordinates. One step in Minkowski's proof amounts to considering a metric on the plane for which the unit ball at any point in R 2 (in fact in Z 2 ) is a translate of the initial convex domain. Such a metric is not Euclidean, it is translation invariant and the straight Euclidean lines are shortest paths. The geometric study of this type of metrics is called (since Hilbert's writings) Minkowski Geometry. We refer to [START_REF] Busemann | The geometry of geodesics[END_REF][START_REF] Day | Normed linear spaces[END_REF][START_REF] Martini | The geometry of Minkowski spaces -a survey. I[END_REF][START_REF] Martini | Recent results in Minkowski geometry[END_REF][START_REF] Thompson | Minkowski geometry. Encyclopedia of Mathematics and its Applications[END_REF] for general expositions of the subject. Minkowski formulated the basic principles of this geometry in his 1896 paper [START_REF] Minkowski | Sur les propriétés des nombres entiers qui sont dérivées de l'intuition de l'espace[END_REF], and these principles are recalled in Hilbert's lecture [START_REF] Hilbert | The English translation was also reprinted in "Mathematical developments arising from Hilbert problems[END_REF] (Problem IV).

An express description of Minkowski geometry is the following: choose a convex set Ω in R n that contains the origin. For points p and q in R n , define a number δ > 0 as follows. First dilate Ω by the factor δ and then translate the set in such a way that 0 is sent to p and q lies on the boundary of the resulting set. In other word δ is defined by the condition q ∈ ∂(p + δ • Ω).

(1.1)

We denote by δ(p, q) the number defined in this way. The function δ : R n ×R n → R + is what we call a weak metric. It satisfies the triangle inequality and δ(p, p) = 0. It is not symmetric in general and it can be degenerate in the sense that δ(p, q) = 0 does not imply p = q. On the other hand, the straight lines are geodesics for this metric and δ is translation invariant. Minkowski geometry is the study of such weak metrics. It plays an important role in convexity theory and in Finsler geometry where Minkowski spaces play the role of flat spaces.

There is a vast literature on Minkowski metrics, and the goal of the present chapter is to provide for the reader of this handbook some of the basic definitions and facts in the theory of weak Minkowski metrics, because of their relation to Hilbert geometry, and to give some examples. We give complete proofs of most of the stated results. We end this chapter with a discussion about the relations and analogies between Minkowski geometry and Funk and Hilbert geometries.

Weak metric spaces

We begin with the definition of a weak metric space. Definition 2.1 (Weak metric). A weak metric on a set X is a map δ : X ×X → [0, ∞] satisfying the following two properties: a.) δ(x, x) = 0 for all x in X; b.) δ(x, y) + δ(y, z) ≥ δ(x, z) for all x, y and z in X.

We often require a weak metric to satisfy somme additional properties. In particular one says that the weak metric δ on X is c.) separating if x = y implies δ(x, y) > 0, d.) weakly separating if x = y implies max {δ(x, y), δ(y, x)} > 0, e.) finite if δ(x, y) < ∞, f.) reversible (or symmetric) if δ(y, x) = δ(x, y), g.) quasi-reversible if δ(y, x) ≤ Cδ(x, y) for some constant C, for all x and y in X.

One sometimes says that δ is strongly separating if condition (b) holds, in order to stress the distinction with condition (d). Observe that for reversible metrics both notions of separation coincide.

A metric in the classical sense is a reversible, finite and separating weak metric. Thus, it satisfies 0 < δ(x, y) = δ(y, x) < ∞ for all x = y ∈ X. Definition 2.2. Let U ⊂ X be a convex subset of a real vector space X. A weak metric δ in U is said to be projective (or projectively flat )metric!projectively flatprojectively flat!metric if satisfies the condition

δ(x, y) + δ(y, z) = δ(x, z) (2.1)
whenever the three points x, y and z in U are aligned and y ∈ [x, z], the affine segment from x to z (equivalently if y = tx + (1t)z for some 0 ≤ t ≤ 1). The weak metric is strictly projective if it is projective and

δ(x, u) + δ(u, z) > δ(x, z) whenever u ∈ [x, z].
Example 2.3. A function ϕ : R n → R is said to be monotone increasing if whenever x and y in R n satisfy x j ≤ y j for 1 ≤ j ≤ n we have ϕ(x) ≤ ϕ(y).

For any collection {ϕ a } a∈A of monotone increasing functions defined on R n , the weak metric defined as

δ(x, y) = sup a∈A |ϕ(y) -ϕ(x)| is projective. A concrete example is given by δ(x, y) = max 1≤j≤n |e yj -e xj |.
Definition 2.4 (Weak Minkowski metric). A weak Minkowski metric on a real vector space X is a weak metric δ on X that is translation invariant and projective.

The weak metric in Example 2.3 is projectively flat, but in general it is not a weak Minkowski metric. Example 2.5. Let X be a real vector space and ϕ : X → R a linear form. Define δ ϕ (x, y) = max{0, ϕ(yx)}. Then δ is a weak Minkowski metric. It is finite, but it is neither reversible nor weakly separating.

In functional analysis, given a real vector space X, the collection of sets

B (ϕ,x,r) = {y ∈ X δϕ(x, y) < r} ⊂ X,
where x ∈ X is an arbitrary point, r > 0 and ϕ ∈ X * is an arbitrary linear form generate a topology which is called the weak topology on X. This observation is a possible justification for the name "weak metric" that we give to such functions. The terminology has its origin in the work of Ribeiro who was interested around 1943 [START_REF] Ribeiro | Sur les espaces à métrique faible[END_REF] in some generalization of the Urysohn metrization theorem for the topology associated to δ.

Example 2.6 (Counterexample). Let X be a real vector space and let : X → R be a norm on X. Then δ(x, y) = max{ yx , 1} is a metric that is translation invariant, but it is not a Minkowski metric because it is not projective. Indeed, suppose z = 1, then

δ(0, 2z) = 1 < δ(0, z) + δ(z, 2z) = 2.
In this example, the metric is "projective for small distances", in the sense that if zx ≤ 1 and y ∈ [x, z], then (2.1) holds. On the other hand, large closed balls are not compact; in fact any ball of radius ≥ 1 is equal to the whole space X.

Example 2.7 (Counterexample). This is a variant of the previous example. Let again : X → R be a norm on the real vector space X. Then ρ α (x, y) = yx α is a metric if and only if 0 < α ≤ 1. It is clearly translation invariant, but it is not projective if α < 1, and thus it is not a Minkowski metric.

Unlike the previous metric δ, the metric ρ α is not projective for small distances (if α < 1). On the other hand, every closed ball is compact.

3 Weak Minkowski norms Proposition 3.1. Let δ be a weak Minkowski metric on some real vector space X and set F (x) = δ(0, x). Then the function F : X → [0, ∞] satisfies the following properties i.)

F (x 1 + x 2 ) ≤ F (x 1 ) + F (x 2 ) for all x 1 , x 2 ∈ X.
ii.) F (λx) = λF (x) for all x ∈ X and for all λ ≥ 0.

Proof. The first property is a consequence of the triangle inequality together with the fact that δ is translation invariant:

F (x + y) = δ(0, x + y) ≤ δ(0, x) + δ(x, x + y) = δ(0, x) + δ(0, y) = F (x) + F (y).
To prove the second property, observe for any x ∈ X and any λ, µ ≥ 0 we have δ(0, λx) + δ(λx, (λ + µ)x) = δ(0, (λ + µ)x), because λx belongs to the segment [0, (λ+µ)x]. Since we have δ(λx, (λ+µ)x) = δ(0, µx) = F (µx), the previous identity can be written as

F (λx) + F (µx) = F ((µ + λ)x) (3.1)
and we conclude from the next lemma that F (λx) = λF (x) for all λ > 0. We also have

F (0 • x) = 0 • F (x) = 0 since F (0) = δ(0, 0) = 0. Lemma 3.2. Let f : R + → [0, ∞] be a function such that f (µ + λ) = f (λ) + f (µ) for any λ, µ ∈ R + , then f (λ) = λf (1)
for every λ > 0.

Proof. We first assume f (a) < ∞ for every a ∈ R + . We have by hypothesis

f (k • a) = f (((k -1) + 1) • a) = f ((k -1) • a) + f (a)
for any k ∈ N. We thus have by induction

f (k • a) = k • f (a)
for any k ∈ N and any a ∈ R + . Using the above identity with k, m ∈ N, we have

m • f k m = f m k m = f (k) = k • f (1).
Dividing this identity by m we obtain f (α) = αf (1) for any α ∈ Q + . Consider now λ ∈ R + arbitrary, and choose

α 1 , α 2 ∈ Q + such that α 1 < λ < α 2 . Then f (λ) = f (α 1 ) + f (λ -α 1 ) > f (α 1 ) = α 1 f (1)
and

f (λ) = f (α 2 ) -f (α 2 -λ) < f (α 2 ) = α 2 f (1).
Since α 2α 1 > 0 is arbitrarily small, we deduce that f (λ) = λf (1) for any λ > 0. So far we have assumed f (a) < ∞ for any a > 0. Assume now there exists a > 0 such that f (a) = ∞. Then f (λ) = ∞ for any λ > 0. Indeed choose an integer k such that kλ > a. Then

kf (λ) = f (kλ) = f (kλ -a) + f (a) ≥ f (a) = ∞. Therefore f (λ) = f (1) = ∞. Definition 3.3. A function F : X → [0, ∞]
defined on a real vector space X is a weak Minkowski norm if the following two conditions hold: i.) F (x 1 + x 2 ) ≤ F (x 1 ) + F (x 2 ) for all x 1 , x 2 ∈ X.

ii.) F (λx) = λF (x) for all x ∈ X and for all λ ≥ 0. Proposition 3.1 states that a weak Minkowski metric determines a weak Minkowski norm. Conversely, a weak Minkowski norm defines a weak Minkowski metric δ F by the formula

δ F (x, y) = F (y -x). (3.2)
We then naturally define a weak Minkowski F norm to be

• separating if x = 0 implies F (x) > 0;
• weakly separating if x = 0 implies max{F (x), F (-x)} > 0;

• finite if F (x) < ∞; • reversible (or symmetric) if F (-x) = F (x);
for any x ∈ X.

Example 3.4. The function

F : R 2 → [0, ∞] defined by F (x 1 , x 2 ) = max{x 1 , 0} if x 2 = 0 and F (x 1 , x 2 ) = ∞ if x 2 =
0 is a weak Minkowski norm which is neither finite, nor separating, nor symmetric. It is however weakly separating.

Observe that in this example F is finite on some vector subspace of R 2 . This is a general fact: Proposition 3.5. Let F : X → [0, ∞] be a weak Minkowski norm on the real vector space X and set D F = {x ∈ X : F (x) < ∞}. Then D F is a vector subspace of X. Furthermore, the restriction of F to any finite-dimensional subspace E ⊂ D F is continuous.

Proof. If x, y ∈ D F , then F (x) and F (y) are finite and therefore

F (x + y) ≤ F (x) + F (y) < ∞ and F (λx) = λF (x) < ∞. Therefore x + y ∈ D F and λx ∈ D F ,
which proves the first assertion.

To prove the second part, we consider a finite-dimensional subspace E ⊂ D F and we choose a basis e 1 , e 2 , . . . , e m ∈ E. Define the constant

C = max 1≤j≤m (F (e j ) + F (-e j )).
For an arbitrary vector x = m j=1 α j e j ∈ E we then have

F (x) ≤ m j=1 F (α j e j ) ≤ C • m j=1 |α j |. In particular if x → 0, then F (x) → 0. More generally, if a sequence x ν ∈ E converges to some a ∈ E, then lim sup ν→∞ F (x ν ) = lim sup ν→∞ F (a + (x ν -a)) ≤ F (a) + lim sup ν→∞ F ((x ν -a)) = F (a). Since F (a) ≤ F (x ν ) + F (a -x ν ) we also have F (a) ≤ lim inf ν→∞ (F (x ν ) + F (a -x ν )) = lim inf ν→∞ F (x ν ). It follows that lim sup ν→∞ F (x ν ) ≤ F (a) ≤ lim inf ν→∞ F (x ν ),
and the continuity on E follows.

Corollary 3.6. Any weak Minkowski norm on a finite-dimensional vector space X is lower semi-continous.

Proof. We need to prove that F (a) ≤ lim inf ν→∞ F (x ν ) for every sequence

x ν ∈ X converging to a. If F (a) = ∞, then a belongs to the open set X \ D F . It follows then that x ν ∈ D F for large enough ν and therefore lim inf ν→∞ F (x ν ) = ∞ = F (a).
If F (a) < ∞, then two cases may occur. If infinitely many x ν belong to D F , then by the previous proposition we have

F (a) = lim ν→∞,xν ∈DF F (x ν ) = lim inf ν→∞ F (x ν ).
If on the other hand D F contains only finitely many elements of the sequence

x ν , then lim inf ν→∞ F (x ν ) = ∞ > F (a).
Definition 3.7. Given a weak Minkowski norm F on a vector space X we define the open and closed unit balls at the origin as

Ω F = {x ∈ X : F (x) < 1}, and Ω F = {x ∈ X : F (x) ≤ 1}.
The set

I F = {x ∈ X : F (x) = 1}
is called the unit sphere or the indicatrix of F .

Proposition 3.8. Let F be a weak Minkowski norm on a finite-dimensional vector space X. Then the following are equivalent

(1) F is finite, (2) F is continuous, (3) Ω F is open, (4) 0 is an interior point of Ω F .
Proof. The implication (1) ⇒ (2) is Proposition 3.5 and the implications (2) ⇒ (3) ⇒ (4) are obvious. To prove (4) ⇒ (1) we suppose F is not finite. Then there exists a ∈ X such that F (a) = ∞. Thus, F (λa) = ∞ for all λ > 0, in particular λa ∈ Ω F for all λ > 0 and therefore 0 is not an interior point of Ω F .

Proposition 3.9. Let F be a weak Minkowski norm on R n . Then the following are equivalent

(1) F is separating (i.e. F (x) > 0 for all x = 0),

(2) F is bounded below on the Euclidean unit sphere S n-1 ⊂ U,

(3) Ω F is bounded.

Proof.

(1) ⇒ (2): Suppose that F is not bounded below on S n-1 . Then there exists a sequence x j ∈ S n-1 such that F (x j ) → 0. Choosing a subsequence if necessary, we may assume, by compactness of the sphere, that F (x j ) < ∞ for all j, i.e. x j ∈ D F ∩S n-1 and that x j converges to some point

x 0 ∈ D F ∩S n-1 .
Since F is continuous on D F , we have F (x 0 ) = lim j→∞ F (x j ) = 0. Since x 0 = 0 (it is a point on the sphere), it follows that F is not separating.

(2) ⇒ (3): Condition (2) states that there exists µ > 0 such that F (x) ≥ µ for all x ∈ S n-1 . Therefore F (y) ≤ 1 implies y ≤ 1 µ . (3) ⇒ (1): Suppose F is non separating. Then there exists x = 0 with F (x) = 0. Therefore F (λx) = 0 for any λ > 0. In particular R + x ⊂ Ω F which is therefore unbounded. Definition 3.10. A Minkowski norm is a weak Minkowski norm that is finite and separating. It is simply called a norm if it is furthermore reversible.

To a finite and separating norm is associated a well-defined topology, viz. the topology associated to the symmetrization of the weak metric defined by Equation (3.2) (which is a genuine metric). For a deeper investigation of various topological questions we refer to the book [START_REF] Cobzas | Functional analysis in asymmetric normed spaces[END_REF] by S. Cobzas.

Corollary 3.11. The topology defined by the distance (3.2) associated to a Minkowski norm on R n coincides with the Euclidean topology.

Proof. Proposition 3.5 implies that F is continuous. From the compactness of the Euclidean unit sphere S n-1 we thus have a constant µ > 0 such that µ ≤ F (x) ≤ 1 µ for all points x on S n-1 . It follows that

µ x ≤ F (x) ≤ 1 µ x (3.3)
for all x ∈ R n and therefore F induces the same topology as the Euclidean norm.

The next result shows how one can reconstruct the weak Minkowski norm from its unit ball. Proposition 3.12. Let Ω ⊂ R n be a convex set containing the origin. Define a function F : R n → [0, ∞] by

F (x) = inf{t ≥ 0 : x ∈ t • Ω}. (3.4)
Then F is a weak Minkowski norm and Ω F coincides with the closure of Ω,

that is Ω = {x ∈ R n : F (x) ≤ 1}. Furthermore, if Ω is open, then Ω = {x ∈ R n : F (x) < 1}.
The function F defined by (3.4) is called the Minkowski functional of Ω.

Proof. We need to verify the two conditions in Definition 3.3. For λ > 0, we have

F (λx) = inf{s ≥ 0 : λx ∈ s • Ω} = inf{s ≥ 0 : x ∈ s λ • Ω} = (s=λt) λ inf{t ≥ 0 : x ∈ t • Ω} = λF (x).
Now because Ω is convex we have for s, t > 0

x s ∈ Ω and y s ∈ Ω =⇒ x + y s + t = s • x s + t • y t s + t ∈ Ω. Therefore F (x) < s and F (y) < t =⇒ F (x + y) < s + t,
which is equivalent to F (x + y) ≤ F (x) + F (y). his proves the first part of the proposition.

To prove the remaining assertions, observe that F (x) ≤ 1 means that tx ∈ Ω for any 0 < t < 1 and thus x ∈ Ω. This shows that

Ω ⊂ {x ∈ R n F (x) ≤ 1} ⊂ Ω.
The converse inclusion Ω ⊂ {x ∈ R n F (x) ≤ 1} follows from the lower semicontinuity of F (Corollary 3.6). Finally, if Ω is open then F is continuous (Proposition 3.8) and therefore Ω = {x ∈ R n : F (x) < 1}.

Thus we have established one-to-one correspondences between weak Minkowski metrics on R n , weak Minkowski norms and closed convex sets containing the origin. The closed convex set associated to a weak Minkowski norm F is the set Ω F = {x ∈ X : F (x) ≤ 1}. The associated weak metric is separating if and only if the associated convex set is bounded and the metric is finite if and only if the origin is an interior point of the convex set. Remark 3.13. These concepts have some important consequences in convex geometry. For instance one can easily prove that every unbounded convex set in R n must contain a ray. Indeed, let Ω ⊂ R n be unbounded and convex. One may assume that Ω contains the origin. Then by Proposition 3.9 its weak Minkowski functional F is not separating, that is, there exists a = 0 in R n such that F (a) = 0; but then F (λa) = 0 for every λ > 0 and therefore the ray Ω contains the ray R + a.

Let us conclude this section with two important results from Minkowski geometry. A Minkowski norm on R n is said to be Euclidean if it is associated to a scalar product. Proposition 3.14. Let δ be a Minkowski metric on R n . Then δ is a Euclidean metric if and only the ball

B (a,r) = {x ∈ X δ(a, x) < r} ⊂ R n
(for some arbitrary a ∈ R n and r > 0) is an ellipsoid centered at a. Notice that the above proposition is false if the ellipsoid is not centered at a.

Proof. Recall that by definition an (open) ellipsoid is a convex set in R n that is the affine image of the open Euclidean unit ball. If the weak metric δ is Euclidean, then it is obvious that every ball is an ellipsoid. Conversely, suppose that some ball of an arbitrary Minkowski metric δ is an ellipsoid. Then the ball with the same radius centered at the origin is also an ellipsoid since δ is translation invariant, that is,

B (0,r) = {x ∈ X : F (x) = δ(0, x) < r} is an ellipsoid. But then Ω = B (0,1) = 1 r • B (0,1)
is also an ellipsoid. Changing coordinates if necessary, one may assume that

Ω = {x ∈ R n i x 2 i < 1},
which is the Euclidean unit ball. It follows that

F (x) = inf{t > 0 x ∈ t Ω} = inf{t > 0 x < t} = x
where • denotes the Euclidean norm.

We have the following result on the isometries of a Minkowski metric.

Theorem 3.15. Let δ be a Minkowski metric on R n . Then every isometry of δ is an affine transformation of R n , and the group Iso(R n , δ) of isometries of δ is conjugate within the affine group to a subgroup of the group E(n) of Euclidean isometries of R n . Furthermore Iso(R n , δ) is conjugate to the full group E(n) if and only if δ is a Euclidean metric.

Proof. The first assertion is the Mazur-Ulam Theorem, see [START_REF] Nica | The Mazur-Ulam theorem[END_REF].. To prove the second assertion, we recall that every bounded convex set Ω in R n with non-empty interior contains a unique ellipsoid J Ω ⊂ Ω of maximal volume. This is called the John ellipsoid of Ω, see [START_REF] Barvinok | A course in convexity[END_REF].

Let us consider the unit ball Ω = B (δ,0,1) of our Minkowski metric and let us denote by J its John ellipsoid and by z ∈ J its center. We call J * = Jz the centered John ellipsoid of Ω. Consider now an arbitrary isometry g ∈ Iso(R n , δ). Set g(x) = g(x)b, where b = g(0). Then g is an isometry for δ fixing the origin. By construction and uniqueness, the centered John ellipsoid is invariant: g(J * ) = J * . There exists an element

A ∈ GL n (R) such that AJ * = B is the Euclidean unit ball. Let us set f := A • g • A -1 . Then f (B) = B.
By Mazur-Ulam g is a linear map, therefore f is a linear map preserving the Euclidean unit ball, which means that f ∈ O(n). We thus obtain

g(x) = A -1 (f (x) + Ab)A
where A is linear and x → f (x) + Ab is a Euclidean isometry.

To prove the last assertion, one may assume, changing coordinates if necessary, that Iso(R n , δ) = E(n). Then the δ-unit ball Ω is invariant under the orthogonal group O(n) and it is therefore a round sphere. We now conclude from Proposition 3.14 that δ is Euclidean. [START_REF] Busemann | The geometry of geodesics[END_REF] The midpoint property Definition 4.1. A weak metric δ on the real vector space X satisfies the midpoint property if for any p, q ∈ X we have

δ(p, m) = δ(m, q) = 1 2 δ(p, q)
where m = 1 2 (p + q) is the affine midpoint of p and q. To describe the main consequence of this property, we shall use the notion of dyadic numbers. Definition 4.2. A dyadic number is a rational number of the type λ = 2 -k m with m, k ∈ Z. We denote the set of dyadic numbers by

D = ∞ k=0 2 -k Z,
and the subset of nonnegative dyadic numbers by D + ⊂ D. Proposition 4.3. Let δ be a weak metric on the real vector space X. Then δ satisfies the midpoint property if and only if for any pair of distinct points p, q ∈ X and for any µ, λ in D with µ ≤ λ, we have

δ(γ(µ), γ(λ)) = (λ -µ) • δ(p, q) (4.1)
where γ(t) = tp + (1t)q.

Proof. It is obvious that if (4.1) holds then δ satisfies the midpoint property. The proof of the other direction requires several steps. Assume that δ satisfies the midpoint property. Then we have

δ(p, γ( 1 2 )) = 1 2 δ(p, q) and δ(p, γ(2) = 2δ(p, q).
By an induction argument we then have

δ(p, γ(2 m )) = 2 m δ(p, q) (4.2)
for any k ∈ N. Because p is the midpoint of γ(-2 m ) and γ(2 m ), we deduce that

δ(γ(-2 m ), γ(2 m )) = 2 m+1 δ(p, q). (4.3) Now we have for k ∈ Z δ(γ(k -1), γ(k)) = δ(γ(k), γ(k + 1))
since γ(k) is the midpoint of γ(k -1) and γ(k + 1). Because δ(p, q) = δ(γ(0), γ(1)), we deduce that δ(γ(k), γ(k + 1)) = δ(p, q), and by the triangle inequality we have δ(γ(i), γ(j)) ≤ (ji)δ(p, q) (4.4)

for any i, j ∈ Z with i < j. We will show that this inequality is in fact an equality. Choose m ∈ N with 2 m ≥ max(|i|, |j|)). Then we have from (4.3) and (4.4)

2 m+1 δ(p, q) = δ(γ(-2 m ), γ(2 m )) ≤ δ(γ(-2 m ), γ(i)) + δ(γ(i), γ(j)) + δ(γ(j), γ(2 m )).
Using now (4.4) we have δ(γ(i), γ(j)) ≤ (ji)δ(p, q), but also

δ(γ(-2 m ), γ(i)) ≤ (i + 2 m )δ(p, q), and
δ(γ(j), γ(2 m )) ≤ (2 m -j)δ(p, q). Since (i + 2 k ) + (j -i) + (2 m -j) = 2 m+1
all the above inequalities must be equalities. Thus, we have established that δ(γ(i), γ(j)) = (ji)δ(p, q) (4.5)

for any i, j ∈ Z.

Let us now fix k ∈ N and set q k = γ(2 -k ) and

γ k (t) = γ(t2 -k ) = tp + (1 -t)q k .
Applying (4.5) to γ k we have

δ(γ k (i), γ k (j)) = (j -i)δ(p, q k ) = (j -i)2 -k δ(p, q).
The latter can be rewritten as

δ(γ( i 2 k ), γ( j 2 k )) = ( j 2 k -i 2
k )δ(p, q) for any i, j ∈ Z and k ∈ N, which is equivalent to (4.1) for any dyadic number µ, λ with µ ≤ λ.

The next result is a generalization to the case of weak metrics of a characterization of Minkowski geometry due to H. Busemann in [4, §17]. Proof. If δ is a Minkowski metric, then it is projective and since δ is finite (by hypothesis), it follows from Propositions 3.1 and 3.5 that the distance is given by

δ(x, y) = F (y -x),
where F is a weak Minkowski norm. The continuity of δ follows now from Proposition 3.5 and the midpoint property follows from property (ii) in Proposition 3.1.

Conversely , let us assume that the weak metric δ satisfies the midpoint property and that it is continuous on every line. We need to show that δ is projective and translation invariant.

We first observe that if a, b ∈ R n are two distinct points with δ(a, b) = 0 and if x and y are two points aligned with a and b such that (y -x) is a nonnegative multiple of (ba), then

δ(x, y) δ(a, b) = |y -x| |b -a| , (4.6)
where |q -p| denotes the Euclidean distance between p and q in R n . This follows from Proposition 4.3, together with the continuity of δ on lines and the density of D in R. This immediately implies that δ(p, z) + δ(z, q) = δ(p, q) whenever z ∈ [p, q], meaning that the weak metric δ is projective.

To prove the translation invariance, we consider four points p, q, p ′ , q ′ with (q ′p ′ ) = (qp). If the four points are on a line, then (4.6) implies that δ(p ′ , q ′ ) = δ(p, q). If the four points are not on a line, then pqq ′ p ′ is a nondegenerate parallelogram. Assume also 0 < δ(p, q) < ∞ and denote by L + pq the ray with origin p through q and L qq ′ the line passing through q and q ′ . Choose a sequence y j ∈ L + pq such that |y j -p| → ∞ and set

x j = L p ′ yj ∩ L + qq ′ . p ′ p q q ′ yj xj We then have 1- δ(p, p ′ ) δ(p, y j ) = δ(p, y j ) -δ(p, p ′ ) δ(p, y j ) ≤ δ(p ′ , y j ) δ(p, y j ) ≤ δ(p ′ , p) + δ(p, y j ) δ(p, y j ) = δ(p ′ , p) δ(p, y j ) +1.
Using (4.6), we have δ(p, y j ) → ∞, therefore

lim j→∞ δ(p ′ , x j ) δ(p, q) = lim j→∞ δ(p ′ , y j ) δ(p, y j ) = 1.
Because x j → q ′ on the line L qq ′ , we have by hypothesiss lim j→∞ δ(x j , q ′ ) = lim j→∞ δ(q ′ , x j ) = 0, and since

δ(p ′ , q ′ ) -δ(x j , q ′ ) ≤ δ(p ′ , x j ) ≤ δ(p ′ , q ′ ) + lim j→∞ δ(q ′ , x j ),
we have δ(p ′ , x j ) → δ(p ′ , q ′ ). Therefore

δ(p ′ , q ′ ) δ(p, q) = lim j→∞ δ(p ′ , x j ) δ(p, q) = 1.
It follows that for a nondegenerate parallelogram p, q, q ′ , p ′ , we have δ(p ′ , q ′ ) = δ(p, q). Suppose now that δ(p, q) = 0. Then we also have δ(p ′ , q ′ ) = 0 for otherwise, exchanging the role of p, q and p ′ , q ′ in the previous argument, we get a contradiction.

We thus have established that in all cases δ(p ′ , q ′ ) = δ(p, q) if q ′p ′ = qp. In other words, δ is translation invariant. Since it is projective, this completes the proof that it is a weak Minkowski metric.

Example 4.5 (Counterexample). Let X a be real vector space and let h :

X → R be an injective Q-linear map. Then the function δ : X × X → R defined by δ(x, y) = |h(x) -h(y)|
is a metric which is translation invariant and satisfies the midpoint property. Yet it is in general not projective (unless h is R-linear, and thus dim R (X) = 1).

Strictly and strongly convex Minkowski norms

Definition 5.1. (i) Let F be a (finite and separating) Minkowski norm in R n with unit ball Ω F . Then F is said to be strictly convex if the indicatrix ∂Ω F contains no non trivial segment, that is, if for any p, q ∈ ∂Ω F , we have [p, q] ⊂ ∂Ω ⇒ p = q.

(ii) The function F is said to be strongly convex if F is smooth on R n \ {0} and the hypersurface ∂Ω F ⊂ R n has everywhere positive Gaussian curvature. Equivalently, the Hessian

g y (η 1 , η 2 ) = 1 2 ∂ 2 ∂u 1 ∂u 2 u1=u2=0 F 2 (y + u 1 η 1 + u 2 η 2 ) (5.1) of F 2 (y) is positive definite for any point y ∈ R n \ {0}.
There are several equivalent definitions of strict convexity in Minkowski spaces, see e.g. [START_REF] Day | Normed linear spaces[END_REF][START_REF] Minkowski | Geometrie der Zahlen[END_REF].

It is clear that a strongly convex Minkowski norm is strictly convex. The converse does not hold: the L p -norm

y p =   n j=1 |y j | p   1/p
is an example of a smooth strictly convex norm which is not strongly convex. Proposition 5.2. Let F be a strongly convex Minkowski norm on R n . Then F can be recovered from its Hessian via the formula

F (y) = g y (y, y) (5.2)
where g y is defined by (5.1).

This result follows from applying twice the following Lemma, which is sometimes called the Euler Lemma. In particular y i ∂ψ

∂y i = 0 if ψ is 0-homogenous.
Recall that a function ψ : R n \ 0 → R is positively homogenous of degree r if ψ(λy) = λ r ψ(y) for all y ∈ R n \ 0 and all λ > 0.

Proof. This is elementary: we just differentiate the function t → ψ(ty) = t r ψ(y) to obtain ∂ψ ∂y i (ty) • y i = rt r-1 • ψ(y),

and set t = 1.
If F is a strongly convex Minkowski norm on R n , then Formula (5.1) defines a Riemannian metric g y on R n \ {0}. Using Lemma 5.3, on gets that g y is invariant under homothety, that is we have g λy = g y for every λ > 0 and y ∈ R n \ {0}. Furthermore F is determined from this metric by Equation (5.2). We conclude from these remarks the following: Proposition 5.4. There is a natural bijection between strongly convex Minkowski norms on R n and Riemannians metric on R n \ {0} which are invariant under homothety.

This observation can be used as a founding stone for Minkowski geometry, see e.g. [START_REF] Varga | Zur Begründung der Minkowskischen Geometrie[END_REF], and it plays a central role in Finsler Geometry.

6 The synthetic viewpoint Definition 2.4 of a weak Minkowski space is based on a real vector space X as a ground space. In fact only the affine structure of that space plays a role and we could equivalently start with a given affine space instead of a vector space.

The synthetic viewpoint is to start with an abstract metric space and to try to give a list of natural conditions implying the given metric space to be Minkowskian. This question, and similar questions for other geometries, has been a central and recurring question in the work H. Busemann, and it is implicit in Hilbert's comments on his Fourth Problem [START_REF] Hilbert | The English translation was also reprinted in "Mathematical developments arising from Hilbert problems[END_REF]. Some answers are given in his book The Geometry of Geodesics [START_REF] Busemann | The geometry of geodesics[END_REF], in that book Busemann introduces the notions of G-spaces and Desarguesian spaces. The goal of this section is to give a short account on this viewpoint. We restrict ourselves to the case of ordinary metric spaces. Definition 6.1 (Busemann G-space). A Busemann G-space is a metric space (X, d), satisfying the following four conditions:

(1) (Menger Convexity) Given distinct points x, y ∈ X , there is a point z ∈ X different from x and y such that d(x, z) + d(z, y) = d(x, y).

(2) (Finite Compactness) Every d-bounded infinite set has an accumulation point.

( A typical example of a Busemann G-space (X, d) is a strongly convex Finsler manifold of class C 2 (and in fact of class C 1,1 by a result of Pogorelov). It follows from the definition that any pair of points in a Busemann G-space (X, d) can be joined by a minimal geodesic and that geodesics are locally unique. It is also known that every G-space is topologically homogeneous and that it is a manifold if its dimension is at most 4. We refer to [START_REF] Berestovskiȋ | Locally G-homogeneous Busemann G-spaces[END_REF] for further results on the topology of G-spaces.

Among G-spaces, Busemann introduced the class of Desarguesian spaces. Definition 6.2 (Desarguesian space). A Desarguesian space is a metric space (X, d) satisfying the following conditions:

(1) (X, d) is a a Busemann G-space.

(2) (X, d) is uniquely geodesic, that is every pair of points can be joined by a unique geodesic.

(3) If the topological dimension1 of X equals 2, then Desargues theorem holds for the family of all geodesics.

(4) If the topological dimension of X is greater than 2, then any triple of points lie in a plane, that is, a two-dimensional subspace of X which is itself a G-space.

The reason for assuming Desargues' property in the 2-dimensional case as an axiom is due to the well known fact from axiomatic geometry that it is possible to construct exotic 2-dimensional objects satisfying the axioms of real projective or affine geometry but which are not isomorphic to RP 2 or R 2 (an example of such exotic object is the Moufang plane); these objects do not satisfy Desargues property. Similar objects do not exist in higher dimension and Desargues property is in fact a theorem in all dimensions ≥ 3. Condition (3) in the above definition could be rephrased as follows: If X is 2-dimensional, then it can be isometrically embedded in a 3-dimensional Desarguesian space. We refer to [START_REF] Busemann | The geometry of geodesics[END_REF] and [START_REF] Papadopoulos | On Hilbert's Fourth Problem, Handbook of Hilbert geometry[END_REF] for further discussion of Desarguesian spaces.

A deep result of Busemann states that a Desarguesian space can be mapped on a real projective space or on a convex domain in a real affine space with a projective metric. More precisely he proved the following Theorem 6.3 (Theorems 13.1 and 14.1 in [START_REF] Busemann | The geometry of geodesics[END_REF]). Given an n-dimensional Desarguesian space (X, d), one of the following condition holds:

(1) Either all the geodesics are topological circles and there is a homeomorphism ϕ : X → RP n that maps every geodesic in X onto a projective line;

(2) or there is a homeomorphism from X onto a convex domain C in R n that maps every geodesic in X onto the intersection of a straight line with C.

Using the notion of Desarguesian space and following Busemann, we now give two purely intrinsic characterizations of finite-dimensional Minkowski spaces among abstract metric spaces. Note that a Minkowski space (X, d) is a G-space if and only if its unit ball is strictly convex. The first result is a converse to that statement. Theorem 6.4 ([4], Theorem 24.1). A metric space (X.d) is isometric to a Minkowski space if and only if it is a Desarguesian space in which the parallel postulate holds and the spheres are stirctly convex.

Observe that in a Desarguesian space there are well defined notions of lines and planes and therefore Euclid's parallel can be formulated. Using Theorem 6.3 and the parallel postulate, we obtain that (X, d) is isometric to R n with a projectively flat metric. To prove the Theorem, Busemann uses the strict convexity of spheres to establish the midpoint property.

The next result we state involves the notion of Busemann zero curvature. Recall that a geodesic metric space is said to have zero curvature in the sense of Busemann, if the distance between the midpoints of two sides of an arbitrary triangle is equal to half the length of the remaining side. Busemann then formulates the following characterization: Theorem 6.5 ([4], Theorem 39.12). A simply connected finite-dimensional G-space of zero curvature is isometric to a Minkowski space.

Busemann came back several times to the problem of characterizing Minkowskian and locally Minkowskian spaces. In his paper with Phadke [START_REF] Busemann | Minkowskian geometry, convexity conditions and the parallel axiom[END_REF], written 25 years after [START_REF] Busemann | The geometry of geodesics[END_REF], he gave sufficient conditions that are more technical but weaker than those of Theorem 6.5.

Comparison and analogies between Minkowski gaometry and Funk and Hilbert geometries

Given a Minkowski metric δ in R n whose unit ball Ω at the origin is open and bounded, the distance between two points is obtained by setting δ(x, x) = 0 for all x in R n and, for x = y,

δ(x, y) = |x -y| |0 -a + |
where | | denotes the Euclidean metric and the point a + is the intersection with ∂Ω of the ray starting at the origin 0 of R n and parallel to the ray R(x, y) from x to y. This formula is equivalent to (1.1) and it suggest an analogy with the formula for the Funk distance in the domain Ω (see Definition 2.1 in the chapter [START_REF] Papadopoulos | A survey of Funk geometry[END_REF] of this volume). It is also in the spirit of the following definition of Busemann ([4], Definition 17.1): A metric d(x, y) in R n is Minkowskian if for the euclidean metric e(x, y) the distances d(x, y) and e(x, y) are proportional on each line.

Minkowski metrics share several important properties of the Funk and the Hilbert metrics, and it is interesting to compare these classes of metrics. Let us quickly review some of the analogies.

We start by recalling that in the formulation of Hilbert's fourth problem which asks for the construction and the study of metrics on subsets of Euclidean (or of projective) space for which the Euclidean segments are geodesics, the Minkowski and Hilbert metrics appear together as the two examples that Hilbert gives (see [START_REF] Hilbert | The English translation was also reprinted in "Mathematical developments arising from Hilbert problems[END_REF] and the chapter [START_REF] Papadopoulos | On Hilbert's Fourth Problem, Handbook of Hilbert geometry[END_REF] in this volume).

A rather simple analogy between the Minkowski and the Funk geometries is that both metrics are uniquely geodesic if and only if their associated convex sets are strictly convex. (Here, the convex set associated to a Minkowski metric is the unit ball centered at the origin. The convex set associated to a Funk metric is the set on which this metric is defined.)

Another analogy between Minkowski and Hilbert geometries is the well known fact that a Minkowski weak metric on R n is Riemannian if and only if the associated convex set is an ellipsoid, see Proposition 3.14. This fact is (at least formally) analogous to the fact that the Hilbert geometry of an open bounded convex subset of R n is Riemannian if and only if the convex set is an ellipsoid (see [START_REF] Kay | The Ptolemaic inequality in Hilbert geometries[END_REF] Proposition 19).

As a further relation between Minkowski and Hilbert geometries, let us recall a result obtained by Nussbaum, de la Harpe, Foertsch and Karlsson. Nussbaum and de la Harpe proved (independently) in [START_REF] Nussbaum | Hilbert's projective metric and iterated nonlinear maps[END_REF] and [START_REF] De La Harpe | On Hilbert's metric for simplices[END_REF] that if Ω ⊂ R n is the interior of the standard n-simplex and if H Ω denotes the associated Hilbert metric, then the metric space (Ω, H Ω ) is isometric to a Minkowski metric space. Foertsch and Karlsson proved the converse in [START_REF] Foertsch | Hilbert metrics and Minkowski norms[END_REF], thus completing the proof of the fact that a bounded open convex subset Ω of R n equipped with its Hilbert metric is isometric to a Minkowski space if and only if Ω is the interior of a simplex.

It should be noted that the result (in both directions) was already known to Busemann since 1967. In their paper [START_REF] Busemann | A general version of Beltrami's theorem in the large[END_REF], p. 313, Busemann and Phadke write the following, concerning the simplex:

The case of general dimension n is most interesting. The (unique) Hilbert geometry possessing a transitive abelian group of motions where the affine segments are the chords (motion means that both distance and chords are preserved) is given by a simplex S, ([5] p. 35). If we realize I [the interior of the simplex] as the first quadrant xi > 0 of an affine coordinate system, the group is given by x ′ i = βixi, βi > 0 [...] m is a Minkowski metric because it is invariant under the translations and we can take the affine segments as chords".

We finally mention the following common characterizations of Minkowski-Funk geometries and of Minkowski-Hilbert geometries: Theorem 7.1 (Busemann [6]). Among noncompact and nonnecessarily symmetric Desarguesian space in which all the right and left spheres of positive radius around any point are compact, the Hilbert and Minkowski geometries are characterized by the property that any isometry between two (distinct or not) geodesics is a projectivity. Theorem 7.2 (Busemann [START_REF] Busemann | Spaces with homothetic spheres[END_REF]). A Desarguesian space in which all the right spheres of positive radius around any point are homothetic is either a Funk space or a Minkowski space.

Theorem 4 . 4 .

 44 A finite weak metric δ on R n is a weak Minkowski metric if and only if it satisfies the midpoint property and if its restriction to every affine line is continuous. More precisely, the latter condition means that if a and b are two points in R n , then for any t 0 ∈ R we have limt→t0 δ(γ(t), b)) = δ(γ(t 0 ), b)) and lim t→t0 δ(a, γ(t)) = δ(a, γ(t 0 )), where γ(t) = ta + (1t)b.

Lemma 5 . 3 .

 53 Let ψ : R \ 0 → R be a positively homogeneous functions of degree r. If ψ is of class C k for some k ≥ 1, then the partial derivatives ∂ψ ∂y i are positively homogenous functions of degree r -1 and r • ψ(y) = n i=1 y i ∂ψ ∂y i .

  ) (Local Extendibility) For every point p ∈ X , there exists r p > 0, such that for any pair of distinct points x, y ∈ X in the open ball B(p, r p ), there is a point z ∈ B(p, r p ) \ {x, y} such that d(x, y) + d(y, z) = d(x, z).(4) (Uniqueness of Extension) Let x, y, z 1 , z 2 be four points in X such that d(x, y) + d(y, z 1 ) = d(x, z 1 ) and d(x, y) + d(y, z 2 ) = d(x, z 2 ). Suppose that d(y, z 1 ) = d(y, z 2 ), then z 1 = z 2 .

On page 46 in[START_REF] Busemann | The geometry of geodesics[END_REF], Busemann states that he is using the Menger-Urysohn notion of dimension, but any reasonable notion of topological dimension is equivalent for a G-space.

The first author is partially supported by the French ANR project FINSLER.


Michael Beisswenger
email: michael.beisswenger@tu-dortmund.de

Thierry Chanier

Isabella Chiari

Maria Ermakova

Maarten Van Gompel

Iris Hendrickx

Axel Herold

Henk Van Den Heuvel

Lothar Lemnitzer
email: lemnitzer@bbaw.de

Angelika Storrer

Michael Beißwenger

Computer-mediated communication in TEI: What lies ahead

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

The social web has brought forth various genres of interpersonal communication (computer-mediated communication, henceforth: cmc) such as chats, discussion forums, wiki talk pages, Twitter, comment and discussion threads on weblogs and social network sites. These genres display linguistic and structural peculiarities which differ both from speech and from written text. Projects that want to build and exchange cmc corpora would greatly benefit from a standard that allows the user to annotate these peculiarities in TEI.

From the perspective of several corpus projects which aim at building and annotating cmc corpora for several European languages, this panel will discuss how the models provided by the TEI encoding framework may be adapted to the special requirements of cmc genres.

The basis of the discussion is a customized TEI schema presented at the TEI conference held in Würzburg 2011 [START_REF] Beißwenger | A TEI Schema for the Representation of Computer-mediated Communication[END_REF] 1 . The panel papers will elaborate on basic features that a TEI standard for cmc resources should include and outline open issues with which further work will have to deal.

The overall goal of the panel is to stimulate the discussion within the TEI community about how a standard for the representation of cmc in TEI should look like and what might be a practical and reasonable way to go about creating such a standard.

In order to push the development of a general standard for the representation of cmc genres and cmc discourse forward, the papers in the panel will present problem overviews for basic issues in representing cmc features in TEI P5 and outline perspectives as well as first suggestions for the treament of these challenges through modifications and expansions of the encoding framework. Starting from these suggestions, the group is planning to work out feature requests and load them onto the TEI projects page on sourceforge.net.

After a general introduction, paper 1 asserts that solutions for the representation of cmc in TEI should be included in the official TEI guidelines and not remain a task that research and corpus projects have to solve using individual customizations. In addition, the paper formulates general requirements a framework for the representation of cmc (in TEI) should comply with as well as specific requirements from several projects which are currently building corpora of cmc discourse for four European languages (German, Dutch, French, and Italian).

Taking into account the requirements outlined in paper 1, paper 2 starts with an overview of existing suggestions for the representation of basic structural and linguistic features of cmc discourse in the TEI framework. It then presents considerations on the following open issues: (1) the modeling of different types of citations in cmc postings; (2) the modeling of hypermedia features (hyperlinks and linking structures, embedded media objects); (3) challenges related to the representation of discourse in multimodal cmc environments in which the participants in one interaction space combine a variety of modalities from written, spoken and non-verbal modes.

Paper 3 examines the issue of metadata. It discusses general requirements for representing metadata of cmc resources and outlines a proposal for representing cmc metadata in the TEI framework.

The panel will include 30 minutes of discussion time (15 minutes each after paper 2 and 3).

Paper 1:

Modeling computer-mediated communication in TEI: requirements and perspectives

This paper reports an ongoing work in a network of corpus projects which aim at building and annotating corpora of computer-mediated communication (cmc)2 and asserts that a framework for the representation of cmc should become a part of the TEI guidelines. It gives an overview of research fields in the Humanities and Computer Sciences which would benefit from the availability of such a representation framework and outlines the basic requirements it will have to comply with: o The schema should provide a general model for the description of the structural and linguistic peculiarities of cmc discourse.

o To be useful for a broad range of application contexts in the Humanities, it should not be designed with a single project in mind but it should take into account the specific requirements of several projects (and genre typologies) in which the creation of annotated cmc resources is of interest.

o In order to be suitable for small data sets which are annotated manually and also for the annotation of big data (e.g., reference corpora in Linguistics, large web corpora in the field of Natural Language Processing), its basic structure should be defined in a way that favours or supports (at least partially) automatic annotation procedures.

o The schema should build on a review of models which already exist in the TEI framework (currently TEI P5) and adapt them to the peculiarities of cmc genres in a reasonable and practical way.

o It should reflect the fact that CMC shares characteristics with written text as well as with spoken conversation while at the same time it is significantly different from both in its textual form and in the mode of production and reception.

o It should allow for an easy (and reversible) anonymization of cmc resources for purposes in which they shall be made available for other researchers (e.g., in the case of reference corpora).

o It should allow for an easy referencing of random samples of the resource (e.g., for citation in scientific publications, didactic materials or dictionary articles).

Since papers 2 and 3 of the panel take into consideration the goals and needs of several projects which are currently dealing with the construction of corpora of cmc discourse in four European languages, paper 1 includes a brief presentation of the four projects and an outline of their project-specific requirements for an annotation schema:

o DeRiK ( Deutsches Referenzkorpus zur internetbasierten Kommunikation ) is a joint project of TU Dortmund University and the Berlin-Brandenburg Academy of Sciences (BBAW) and the Humanities which is building a reference corpus of German cmc discourse including the most prominent cmc genres. The DeRiK corpus will form a new component of the reference corpora of contemporary written German collected in the BBAW project Digitales Wörterbuch der deutschen Sprache (DWDS). On the one hand, it is designed as a resource for corpus-based linguistic analyses of language use in German cmc as well as in combination with the DWDS corpus of the impact of cmc genres on contemporary written German. On the other hand it will serve as a resource for the lexicographic description of netspeak vocabulary and cmc-specific processes of lexical-semantic change in the dictionary component of the DWDS online lexical information system3 (cf. [START_REF] Beißwenger | DeRiK: A German Reference Corpus of Computer-Mediated Communication[END_REF]. For annotation, DeRiK is currently using the customized TEI schema for cmc described in [START_REF] Beißwenger | A TEI Schema for the Representation of Computer-mediated Communication[END_REF]. The schema comprises, among others, an element for the description of user contributions to cmc conversations (the divLike element posting), a distinction of two major types of cmc macrostructures (the cmc-specific division types thread and logfile ), a component for modeling the authors of cmc postings as well as elements for the annotation of selected netspeak features in individual user postings (emoticons, interaction words, interaction templates, addressing terms).

o The Dutch reference corpus SoNaR was intended to serve as a general reference for studies involving language and language use. The corpus should provide a balanced account of the standard language and the variation that occurs within it. In doing so, it allows researchers investigating language use in a particular domain (e.g. medicine)

or register (e.g. academic writing) or by a specific group (e.g. professional translators) to relate their data and findings to the general reference corpus. The corpus was also intended to play a role in the benchmarking of tools and annotations. Collected in 2008-2012 the corpus contains 500 Mwords, including discussion lists, e-magazines, websites, Wikipedia, SMS, chats and tweets. SoNaR is delivered in the FoLiA format (van Gompel 2012). FoLiA aims to support a wide variety of linguistic annotations in a generic paradigm and has been successfully adopted by various projects in The Netherlands. To provide support for new media, a type of structure annotation called "event annotation" was added, which fits nicely in the paradigm. SoNaR incorporates support for tweets, chat logs and SMS. The former two have been encoded as events, in which each tweet or chat message constitutes an event. Within the event structure, further subdivisions can optionally be made, such as paragraphs, sentences, words (in case of tokenized data). Elements in FoLiA carry a class from a certain set. In this way flexibility is provided to the user. The sets can be formally defined. The events in SoNaR are assigned classes such as "tweet" or "chatmessage". The actors of the set are also explicitly annotated, and further metadata on the annotation is also supported.

o LETEC ( Learning & Teaching Corpora ). Mulce repository4 is a databank of LETEC corpora built upon online learning situations [START_REF] Reffay | Multimodal Learning and Teaching Corpora Exchange: Lessons learned in 5 years by the Mulce project. Special Issue on dataTEL: Datasets and Data Supported Learning in Technology-Enhanced Learning[END_REF]. All interactions among participants have been collected and structured before their analysis. It assembles a large variety of cmc types: email, forums, chat, blogs, 3D environments with audio and text chats, etc. One of the main components of its XML structure (Mulce-struct)5 is the workspace. It includes descriptions of its members as references to the participants registered in the learning activity, starting and ending dates, the tools and the interaction tracks or acts that occurred using these tools.

Each cmc tool has a detailed and specific structure. Large subparts of the LETEC databank will be integrated in 2013-14 into a nationwide cmc corpus in French where other cmc types, such as SMS, tweets, Wikipedia forums, will be added. The cmc SIG group leading the project belongs to the national consortium IR corpus-écrits in charge of building a reference corpus in French. The cmc SIG has designed a working package which will take care of the cmc TEI structure6 of the whole corpus and work jointly with the European colleagues gathered in this panel.

o Web2Corpus_it ( Corpus italiano di comunicazione mediata dal computer ) is a project funded by Sapienza University of Rome in 2010 aimed at investigating meaning negotiation strategies in cmc. It focuses on conversational, interactive, public, written communication in order to build a genre-balanced cmc corpus of Italian language to be investigated both qualitatively and quantitatively. The genres included are: forum, blog, newsgroup, social network and chat (cf. Chiari and Canzonetti, in press). 7 The collected corpus comprises one million words and has been fully anonymized (by masking), in order to avoid personal details of participants being disclosed, and xml-annotated both for macro-structural properties (thread, post, sender details -avatar | signature | nickname | senderplace subject, date, time, links and embedded media, web action elements and cmc-specific emoticons and tags and addressing terms). At present the corpus is being processed linguistically with a statistical POS tagger and lemmatizer, including a reference machine dictionary (Common Lexicon of Italian) developed in order to include cmc specific lexical items, and will be subsequently manually checked and is planned to be released in late 2013.

These four corpus projects will provide the test bed for an evaluation of the models under construction with cmc discourse from different languages.

Paper 2:

Expanding the TEI encoding framework to genres of computer-mediated communication: considerations and suggestions

The first section of this paper presents some basic suggestions for the expansion of the TEI encoding framework to the structural and linguistic particularities of cmc genres. It takes into account the general requirements as well as the project-specific requirements outlined in paper 1 and builds on the customized TEI schema for cmc which has been presented at the 2011 TEI members meeting (published in [START_REF] Beißwenger | A TEI Schema for the Representation of Computer-mediated Communication[END_REF]). The suggestions describe features for the modeling of corpus documents with stored discourse from cmc genres such as online forums, chats, wiki talk pages, Twitter, weblogs or social network sites and (amongst others) refer to the following basic issues in the description of cmc:

o the representation of user postings in written cmc as units which share characteristics with both text and conversations: under aspects of planning and coherence, they are designed as moves in an ongoing conversation; under the aspect of production and reception they behave just like texts, which first have to be produced and then are presented to and received by the addressee(s) en bloc;

o the need for models for the representation of cmc macrostructures (= the way how series of user postings are grouped / presented to the users, e.g., in the form of logfiles, different types of threads, timelines etc.); o the need for elements for the annotation of cmc-specific structural and linguistic features on the microlevel of cmc discourse (= the content of the postings which comprises e.g. typical netspeak phenomena such as emoticons, action words, addressing terms; hashtags; speedwriting phenomena, phenomena of nonstandardized writing; embedded hyperlinks and media objects etc.);

With the help of examples from the corpus projects introduced in paper 1, the second section of the paper will offer problem sketches of the following open issues in modeling cmc and outline some first ideas for their treatment in TEI:

o Handling citations: Especially in forums and Bulletin Boards, cmc postings often contain (simple and nested) citations which reproduce content that has originally been part of other authors' prior postings. A schema for the representation of cmc should include a model for the annotation of citations and for referencing citations with the cited prior postings and their authors.

o Cmc data as hyperlinked data: Many cmc resources contain hyperlinks and linking structures. A framework for the representation of cmc interactions must include models for the description of how postings are linked with each other and/or with other interaction-external resources on the internet. In some cmc applications (e.g., microblogging sites such as Twitter) the method of displaying one and the same user posting as part of a sequence may vary depending on the user s choice (cf. e.g. on Twitter the timeline of one author s tweets vs. the timeline of tweets by different authors which include the same hashtag). A general model for cmc resources must provide features for the description of these kinds of structures and of the target sources of the hyperlinks.

o Dealing with data from multimodal cmc environments: In some cmc environments users are communicating not only in a text-based mode but using a combination of text-, audio-, video-and/or 3D-based modalities of interaction (e.g., e-learning platforms, Skype, gaming environments, virtual worlds etc.). One of the challenges related to the representation of cmc discourse recorded in environments of that kind is that contributions created and sent in one modality may contribute to, and indeed supplement, a contribution in another modality. In audio-graphic conferencing environments such as Skype, written postings sent via chat may contribute to an ongoing spoken conversation in the audio modality. In collaborative writing environments, written postings in the chat may contribute to the creation of a longer stretch of text in the word-processing modality. One challenge of treating cmc discourse of that kind is thus the necessity to integrate and align user contributions made in different modalities into a representation of the overall multimodal interaction.

Since TEI provides modules not only for written but also for (transcriptions of) spoken discourse, the different modes could be represented separately (using different TEI modules) while the alignment of the utterances and postings in the different modalities would have to be solved in an additional representation which is connected with the different resources.

Paper 3:

Metadata for cmc documents

Axel Herold 1 , Lothar Lemnitzer 1 , Michael Beißwenger 2 , Isabella Chiari 3

1 Berlin-Brandenburg Academy of Sciences and the Humanities (D) 2 TU Dortmund University (D) 3 Università La Sapienza , Rome (IT)

Extensive and correct metadata has been recognized to be a crucial property of every data object that is used as a primary data source in research contexts. Fine grained metadata allow for identification, location and management of resources (e.g., NISO, 2004) but also provide researchers with crucial information regarding the suitability of a given resource for their particular research interest. The TEI header recognizes all of these metadata requirements to different degrees [START_REF] Burnard | Metadata for corpus work[END_REF].

Our paper will have a strong focus on the encoding of intrinsic properties of different cmc data sets, thus addressing the issue of finding resources which are suitable for a given research question. Ideally, this part of the metadata description is based on the model representing the primary data. In this respect our paper strongly relies on paper 2, which will propose such a model for cmc data.

An example of cmc-specific data types are emoticons: small iconic representations of an interlocutor s emotion or his/her attitude towards an utterance (either self produced or produced by others) or towards a communication peer, to name just some of their communicative functions. It is therefore desirable to either encode normalization and classification schemes for those entities within the metadata description or to provide pointers to such schemes in addition to a suitable markup of these entities within the primary data.

Cmc data often contains large portions of verbosely cited material from previous parts of the discourse. This creates a challenge to the measurement of the extent of a given resource. Depending on the assumed discourse status of cited (parts) of utterances it may be necessary to include or exclude cited material. This is a theory-dependent decision, and it should therefore be possible to give concurrent values for a single unit of measurement. Moreover, metadata information on (the handling of) citations may to some extent be derived from the primary data directly (see paper 2 for handling of citations in primary text).

Distinct typologies for cmc tools (including tools that were used to access the primary data) and cmc genres are needed to account for the broad range of different data sources, e.g., online forums, chats, wikis, Twitter, weblogs, social network sites, learning environments and others. We will suggest mechanisms of referencing a particular typology of cmc genres from within the metadata, however, without making any regulations on which kind of typology should be used and referenced in a given project. Special care must be taken in the metadata description of information about discourse participants to ensure privacy and/or anonymity of the speakers involved in the discourse. Moreover, specific metadata for cmc should also have the function of restoring context information about features of the communication mode of production and reception of cmc texts that are not evident in the text itself. This involves features such as the temporal structuring of the discourse (synchronous vs. asynchronous mode), conversational hierarchies among discourse participants (e. g. blog author vs. commentator), discourse topic/domain or accessibility of the discourse (e. g. private vs. closed vs. public). The availability of social and other context information varies greatly, not only in quantity but also in its quality, according to the primary data source. Therefore a cmc metadata scheme will have to account for different levels of reliability for such information.

Considering the given fourfold structure of the TEI header (file description, encoding description, text profile and revision description), we will identify and discuss different possibilities for recording metadata properties that are specific for cmc data:

• Cmc data comprise properties found in traditional written resources (such as books or newspapers) as well as properties found in resources of (transcribed) spoken language. Both types of resources have previously been provided with TEI-based metadata. Properties shared across different resource types can be expected to be reusable for cmc metadata, e.g., listPerson to denote discourse participants or profileDesc to describe general discourse settings.

• Some metadata properties that cannot be readily encoded using specific elements can still be recorded using the generic feature structure representation (fs). Embedding of feature structures is currently allowed for a limited set of header elements in the TEI such as classCode, extent, language, scriptNote and typeNote. Exploiting the semantic linking mechanism provided by att.datcat (via the ISOcat data category registry; note that classCode provides a native semantic interface via @scheme as well) would allow tailor-made semantics for the properties encoded in such a way. But obviously this adds a level of indirection and does not capture these properties within the TEI directly.

• A third possibility lies in the adaptation of the TEI element inventory or of suggested cmc-specific value sets for existing elements. For individual projects this can already be achieved by TEI customizations but it may hinder interoperability across resources using elements not found in the TEI guidelines which is another argument for why models for the representation of cmc data in TEI should better be part of the official guidelines and not be something that each project needs to solve individually.

We will conclude the paper with a proposed metadata header for TEI documents encoding cmc data. We will also at least for some prominent features of metadata for cmc documents show how the TEI header metadata are related to, and can be converted to, metadata components within the emerging CLARIN Metadata Framework (Component Metadata Infrastructure, CMDI).

The ODD document can be found at http://www.empirikom.net/bin/view/Themen/CmcTEI

Michael Beißwenger 1 , Thierry Chanier 2 , Isabella Chiari 3 , Maria Ermakova 4 , Lothar Lemnitzer 4 , Angelika Storrer 1 , Maarten van Gompel 5 , Henk van den Heuvel 5 1 TU Dortmund University (D) 2 Université Blaise Pascal, Clermont-Ferrand (F) 3 Università La Sapienza , Rome (IT) 4 Berlin-Brandenburg Academy of Sciences and the Humanities (D) 5 Radboud University Nijmegen (NL)

http://wiki.itmc.tu-dortmund.de/cmc/

http://www.dwds.de

http://repository.mulce.org

Schema for the instantiation component of a LETEC corpus. http://lrl-diffusion.univbpclermont.fr/mulce/metadata/mce-schemas/mce_sid.xsd

6 https://groupes.renater.fr/wiki/corpus-ecrits-nouvcom/public/proj-tei/index

http://www.glottoweb.org/web2corpus/