Cormac Walsh

HILBERT AND THOMPSON GEOMETRIES ISOMETRIC TO INFINITE-DIMENSIONAL BANACH SPACES

We study the horofunction boundaries of Hilbert and Thompson geometries, and of Banach spaces, in arbitrary dimension. By comparing the boundaries of these spaces, we show that the only Hilbert and Thompson geometries that are isometric to Banach spaces are the ones defined on the cone of positive continuous functions on a compact space.

Introduction

It was observed by Nussbaum [13, pages 22-23] and de la Harpe [START_REF] De | On Hilbert's metric for simplices[END_REF] that the Hilbert geometry on a finite-dimensional simplex is isometric to a normed space. Later, Foertsch and Karlsson [START_REF] Foertsch | Hilbert metrics and Minkowski norms[END_REF] proved the converse, that is, if a Hilbert geometry on a finite-dimensional convex domain is isometric to a normed space, then the domain is a simplex.

In this paper, we extend this result to infinite dimension. The natural setting is that of order units spaces. An order unit space is a triple (X, C, u) consisting of a vector space X, an Archimedean convex cone C in X, and an order unit u. Let C be the interior of C with respect to the topology on X coming from the order unit norm. Define, for each x and y in C, M(x, y) := inf{λ > 0 | x ≤ λy}.

Since every element of C is an order unit, this quantity is finite. Hilbert's projective metric is defined to be d H (x, y) := log M(x, y)M(y, x), for each x, y ∈ C.

It satisfies d H (λx, νy) = d H (x, y), for all x, y ∈ C and λ, ν > 0, and is a metric on the projective space P (C) of the cone.

In infinite dimension the role of the simplex will be played by the cone C + (K) of positive continuous functions on a compact topological space K. This cone lives in the linear space C(K) of continuous functions on K, and is the interior of C + (K) the cone of non-negative continuous functions on K. The triple (C(K), C + (K), u) forms an order unit space, where u is the function on K that is identically 1.

It is not hard to show that the Hilbert metric on C + (K) is the following:

d H (x, y) = log sup k,k ∈K x(k) y(k) y(k ) x(k ) , for x, y ∈ C + (K),
The map log : C + (K) → C(K) that takes the logarithm coordinate-wise is an isometry when C(K) is equipped with the semi-norm

||z|| H := sup k∈K z k -inf k∈K z k .
Date: December 31, 2015.

1

Denote by ≡ the equivalence relation on C(K) where two functions are equivalent if they differ by a constant, that it, x ≡ y if x = y + c for some constant c. The seminorm ||x|| H is a norm on the quotient space C(K)/≡. This space is a Banach space, and we denote it by H(K). The coordinate-wise logarithm map induces an isometry from the projective space P (C(K)) to H(K).

We show that every Hilbert geometry isometric to a Banach space arises in this way. When we talk about the Hilbert geometry on a cone C, we are assuming that C is the interior of the cone of an order unit space.

Theorem 12.6. If a Hilbert geometry on cone C is isometric to a Banach space, then C is linearly isomorphic to C + (K) for some compact Hausdorff space K.

We also prove a similar result for another metric related to the Hilbert metric, the Thompson metric. This is defined to be d T (x, y) := log max M(x, y), log M(x, y) , for each x, y ∈ C.

Theorem 11.3. If a Thompson geometry on a cone C is isometric to a Banach space, then C is linearly isomorphic to C + (K), for some compact Hausdorff space K.

The main technique we use in both cases is to compare the horofunction boundary of the Banach space with that of the Hilbert or Thompson geometry. The horofunction boundary was first introduced by Gromov [START_REF] Gromov | Hyperbolic manifolds, groups and actions[END_REF]. Since it is defined purely in terms of the metric structure, it is useful for studying isometries of metric spaces.

In finite dimension, the horofunction boundary of normed spaces was studied in [START_REF] Walsh | The horofunction boundary of finite-dimensional normed spaces[END_REF] and that of Hilbert geometries was studied in [START_REF] Walsh | The horofunction boundary of the Hilbert geometry[END_REF]. The results for the Hilbert geometry were used to study the isometry group of this geometry in the polyhedral case [START_REF] Lemmens | Isometries of polyhedral Hilbert geometries[END_REF] and more generally [START_REF] Walsh | Gauge-reversing maps on cones, and Hilbert and Thompson isometries[END_REF]. See [START_REF] Walsh | The horofunction boundary and isometry group of the Hilbert geometry[END_REF] for a survey of these results.

Usually, when one develops the theory of the horofunction boundary, one makes the assumption that the space is proper, that is, that closed balls are compact. For normed spaces and Hilbert geometries, this is equivalent to the dimension being finite. To deal with infinite dimensional spaces, we are forced to extend the framework. For example, we must use nets rather than sequences. In section 2, we reprove some basic results concerning the horofunction boundary in this setting. We study the boundary of normed spaces in section 3. In this and later sections, We make extensive use of the theory of affine functions on a compact set, including some Choquet Theory. To demonstrate the usefulness of the horofunction boundary, we give a short proof of the Masur-Ulam theorem in section 4. As an example, we determine explicitly the Busemann points in the boundary of the important Banach space (C(K), || • || ∞ ) in section 5. Important to our method will be to consider the Hilbert and Thompson metrics as symmetrisations of a non-symmetric metric, the Funk metric. In sections 6, 7, and 8, we study the boundaries of, respectively, the reverse of the Funk metric, the Funk itself, and Hilbert metric. Again, we take a closer look in section 9 at an example, here the cone C + (K). We study the boundary of the Thompson geometry in section 10, which allows us to prove Theorem 11.3 in section 11. We prove Theorem 12.6 in section 12.

Acknowledgements. I had many very useful discussions with Bas Lemmens about this work. This work was partially supported by the ANR 'Finsler'.

Preliminaries

2.1. Hilbert's metric. Let C be cone in a real vector space X. In other words, C is closed under addition and multiplication by non-negative real numbers, and C ∩ -C = 0. The cone C induces a partial ordering ≤ on X by x ≤ y if y -x ∈ C. We say that C is Archimedean if, whenever x ∈ X and y ∈ C satisfy nx ≤ y for all n ∈ N, we have x ≤ 0. An order unit is an element u ∈ C such that for each x ∈ X there is some λ > 0 such that x ≤ λu. An order unit space (X, C, u) is a vector space X equipped with a cone C containing an order unit u.

We define the order unit norm on X:

||x|| u := inf{λ > 0 | -λu ≤ x ≤ λu}.
We use on X the topology induced by || • || u . It is known that, under this topology, C is closed [START_REF] Aliprantis | Cones and duality[END_REF]Theorem 2.55] and has non-empty interior. Indeed, the interior of C is precisely the set of its order units; see [START_REF] Lemmens | Isometries of infinite dimensional Hilbert geometries[END_REF].

On C, we define Hilbert's projective metric as in the introduction.

Hilbert originally defined his metric on bounded open convex sets. Suppose D is such a set, and that we are given two distinct points x and y in D. Define w and z to be the points in the boundary ∂D of D such that w, x, y, and z are collinear and arranged in this order along the line in which they lie. Hilbert defined the distance between x and y to be the logarithm of the cross ratio of these four points: In the case of an infinite-dimensional cone, one may recover Hilbert's original definition if the cone has a strictly positive state, that is, if there exists a continuous linear functional ψ that is positive everywhere on C. In this situation, Hilbert's definition applied to the cross section {x ∈ C | ψ(x) = 1} agrees with the definition in the introduction. It was shown in [START_REF] Lemmens | Isometries of infinite dimensional Hilbert geometries[END_REF] that Hilbert's definition makes sense if and only if the convex domain is affinely isomorphic to a cross section of the cone of an order unit space.

Not every order unit space has a strictly positive state however. For example, take X := R D , the space of real-valued bounded functions on some uncountable set D. The subset of these functions that are non-negative is an Archimedean cone, and the function that is identically 1 is an order unit. On spaces such as these, the metric d H is well-defined even though Hilbert's original construction is not.

2.2. The Funk and reverse-Funk metrics. Essential to our method will be to consider the Hilbert and Thompson metrics as symmetrisations of the Funk metric, which is defined as follows:

d F (x, y) := log M C (x, y),
for all x ∈ X and y ∈ C.

This metric first appeared in [START_REF] Funk | Über Geometrien, bei denen die Geraden die Kürzesten sind[END_REF]. We call its reverse d R (x, y) := d F (y, x) the reverse-Funk metric. Like Hilbert's metric, the Funk metric was first defined on bounded open convex sets. On a cross section D of a cone C, one can show that

d F (x, y) = log |xz| |yz| and d R (x, y) = log |wy| |wx| ,
for all x, y ∈ D. Here w and z are the points of the boundary ∂D shown in Figure 1.

On D, the Funk metric is a quasi-metric, in other words, it satisfies the usual metric space axioms except that of symmetry. On C, it satisfies the triangle inequality but is not non-negative. It has the following homogeneity property:

d F (αx, βy) = d F (x, y) + log α -log β,
for all x, y ∈ C and α, β > 0.

Observe that both the Hilbert and Thompson metrics are symmetrisations of the Funk metric: for all x, y ∈ C,

d H (x, y) = d F (x, y) + d R (x, y) and d T (x, y) = max d F (x, y), d R (x, y) .
2.3. The horofunction boundary. Let (X, d) be a metric space. Associate to each point z ∈ X the function ψ z : X → R,

ψ z (x) := d(x, z) -d(b, z),
where b ∈ X is some base-point. It can be shown that the map ψ : X → C(X), z → ψ z is injective and continuous. Here, C(X) is the space of continuous real-valued functions on X, with the topology of uniform convergence on compact sets. We identify X with its image under ψ.

Let cl denote the topological closure operator. Since elements of cl ψ(X) are equi-Lipschitzian, uniform convergence on compact sets is equivalent to pointwise convergence, by the Ascoli-Arzelà theorem. Also, from the same theorem, the set cl ψ(X) is compact. We call it the horofunction compactification. We define the horofunction boundary of (X, d) to be

X(∞) := cl ψ(X) \ψ(X),
The elements of this set are the horofunctions of (X, d).

Although this definition appears to depend on the choice of base-point, one may verify that horofunction boundaries coming from different base-points are homeomorphic, and that corresponding horofunctions differ only by an additive constant.

In the finite-dimensional setting one commonly considers geodesics parameterised by Z or R. In infinite dimension, however, one must use nets. Recall that a directed set is a nonempty pre-ordered set such that every pair of elements has an upper bound in the set, and that a net in a topological space is a function from a directed set to the space. Definition 2.1. A net of real-valued functions f α is almost non-decreasing if, for any ε > 0, there exists A such that f α ≤ f α + ε, for all α and α greater than A, with α < α .

An almost non-increasing net is defined similarly. Definition 2.2. A net of real-valued functions f α is almost non-increasing if, for any ε > 0, there exists A such that f α ≥ f α -ε, for all α and α greater than A, with α < α .

Observe that if f α is an almost non-increasing net of functions, and m α is a net (on the same directed set) of real numbers converging to zero, then f α + m α is also almost non-increasing.

Definition 2.3. A net in a metric space is almost geodesic if, for all ε > 0, d(b, z α ) ≥ d(b, z α ) + d(z α , z α ) -ε,
for α and α large enough, with α < α . This definition is similar to Rieffel's [START_REF] Rieffel | Group C * -algebras as compact quantum metric spaces[END_REF], except that here we use nets rather than sequences and the almost geodesics are unparameterised.

Proposition 2.4. Let z α be a net in a metric space. Then, z α is an almost geodesic if and only if

ϕ zα := d(•, z α ) -d(b, z α ) is an almost non-increasing net.
Proof. Let ε > 0 be given. Assume z α is almost geodesic. So, for α and α large enough, with α < α ,

d(b, z α ) ≥ d(b, z α ) + d(z α , z α ) -ε.
Let x be a point in the metric space. Combining the above inequality with the triangle inequality concerning the points x, z α , and z α , we get

d(x, z α ) -d(b, z α ) ≥ d(x, z α ) -d(b, z α ) -ε. (1)
We conclude that the net ϕ zα is almost non-increasing. Now assume that ϕ zα is almost non-increasing, in other words that (1) holds when α and α are large enough, with α < α , for all points x. Taking x equal to z α , we get that z α is an almost geodesic.

Proposition 2.5. Let (X, d) be a metric space. Let z α and z α be an almost geodesics in X converging to the same Busemann point. Then, there exists almost-geodesics x β , x β , and y β such that x β is a subnet of both z α and y β , and x β is a subnet of both z α and y β .

Proof. Denote by ξ the common limit of z α and z α , and by D and D the directed sets of z α and z α , respectively, which we may consider to be disjoint. Let N be the set of neighbourhoods of ξ in the horofunction compactification of X.

To be able to handle the two nets simultaneously, we write zα := z α when α ∈ D, and zα := z α when α ∈ D .

Define Y be the set of elements (W, α, ε

) of N × (D ∪ D ) × (0, ∞) such that zα ∈ W and 0 ≤ d(b, zα ) + ξ(z α ) < ε. (2)
We may write Y := E ∪ E , where E is the set of elements (W, α, ε) of Y where α ∈ D and E is the set where α ∈ D .

Define on this set an order by (W, α, ε) ≤ (W , α , ε ) if either the two elements are identical, or if W ⊃ W , ε ≥ ε , and

|d(z α , y) -d(b, y) -ξ(z α )| < ε, for all y ∈ W . (3)
We further require that α ≤ α if α and α either both belong to E or both belong to E . It is not hard to verify that this order is reflexive and transitive.

To show that every pair of elements (W, α, ε) and (W , α , ε ) has an upper bound, take ε := min(ε, ε ), and choose W to be a neighbourhood of ξ smaller than both W and W , small enough that (3) holds for both zα and zα , for all y ∈ W . Finally, α must be chosen large enough in D ∪ D that zα ∈ W and (2) holds.

Define the net y β := zα , for all β := (W, α, ε) in Y. Suppose some λ > 0 is given. If β := (W, α, ε) ∈ Y is large enough that ε < λ/2, and β := (W , α , ε ) ∈ Y is such that β ≤ β , then combining 2 and (3) we get d(b, y β ) + d(y β , y β ) -d(b, y β ) < λ. This proves that the net y β is an almost geodesic.

We define the nets

x β := zα , for β := (W, α, ε) in E, and
x β := zα , for β := (W, α, ε) in E .
These two nets are clearly subnets of y β since both E and E are cofinal subsets of Y. As subnets of an almost geodesic, they are themselves almost geodesics.

For β := (W, α, ε) in E, we may write x β = z a(β) , where a(β) := α.

Proposition 2.6. Let z α be an almost geodesic in a complete metric space (X, d) with basepoint b. If d(b, z α ) is bounded, then z α converges to a point in X.

Proof. For any ε > 0, we have, for α and α large enough with α < α ,

d(b, z α ) ≤ d(b, z α ) + d(z α , z α ) ≤ d(b, z α ) + ε.
Hence, d(b, z α ) is an almost non-decreasing net of real numbers. By assumption, it is bounded above. We deduce that it converges to some real number, as α tends to infinity. But again, for any ε > 0, we have, for α and α large enough with α < α , d(z α , z α ) ≤ d(b, z α ) -d(b, z α ) + ε. So, we see that z α is a Cauchy net, and hence converges since X is assumed complete.

2.4. The detour cost. Let (X, d) be a metric space with base-point b. One defines the detour cost for any two horofunctions ξ and η in X(∞) to be

H(ξ, η) := sup W ξ inf x∈W ∩X d(b, x) + η(x) ,
where the supremum is taken over all neighbourhoods W of ξ in X ∪ X(∞). This concept first appeared in [START_REF] Akian | The max-plus Martin boundary[END_REF] in a slightly different setting. More detail about it can be found in [START_REF] Walsh | The horoboundary and isometry group of Thurston's Lipschitz metric[END_REF].

Lemma 2.7. Let ξ and η be horofunctions of a metric space (X, d). Then, there exists a net converging to ξ such that

H(ξ, η) = lim α d(b, z α ) + η(z α ) .
Proof. To ease notation, write f (x) := d(b, x) + η(x), for all x ∈ X.

Let N be the set of neighbourhoods of ξ in X ∪ X(∞). Define an order on the set We can then take x ∈ W ∩ X such that f (x) ∈ E. So, α := (W, x) satisfies α ≥ (W , x ) and f (z α ) ∈ E. This shows that H(ξ, η) is a cluster point of the net f (z α ).

D := {(W, x) ∈ N × X | x ∈ W ∩ X} by (W 1 , x 1 ) ≤ (W 2 , x 2 ) if W 1 ⊃ W 2 .
Therefore, there is some subnet w β of z α such that f (w β ) converges to H(ξ, η).

The following was proved in [START_REF] Walsh | Minimum representing measures in idempotent analysis[END_REF]Lemma 3.3] in a slightly different setting. There is a proof in [START_REF] Walsh | The horoboundary and isometry group of Thurston's Lipschitz metric[END_REF] that works with very little modification in the present setting with nets.

Lemma 2.8. Let z α be an almost-geodesic net converging to a Busemann point ξ, and let y ∈ X. Then, lim

α d(y, z α ) + ξ(z α ) = ξ(y).
Moreover, for any horofunction η,

H(ξ, η) = lim α d(b, z α ) + η(z α ) .
Theorem 2.9. A horofunction ξ is a Busemann point if and only if H(ξ, ξ) = 0.

Proof. If ξ is a Busemann point, then it follows from Lemma 2.8 that H(ξ, ξ) = 0. Now assume that ξ is a horofunction satisfying H(ξ, ξ) = 0. By Lemma 2.7, there is a net z α : D → X in the metric space (X, d) converging to ξ such that d(b, z α ) + ξ(z α ) converges to zero.

Define the set

D := (α, β, ε) ∈ D × D × R | ε > 0, α ≤ β, |d(z α , z γ ) -d(b, z γ ) -ξ(z α )| < ε for all γ ≥ β
Observe that, for any α ∈ D and ε > 0, there exists β ∈ D such that (α, β, ε) ∈ D , because d(•, z γ ) -d(b, z γ ) converges to ξ(•) pointwise. Define on D the order relation ≤, where (α, β, ε) ≤ (α , β , ε ) if either (α, β, ε) and (α , β , ε ) are identical, or if β ≤ α and ε ≥ ε . This relation is easily seen to be reflexive and transitive. Also, it is not hard to show that D is directed by ≤.

The map h : (α, β, ε) → α is monotone, and its image is cofinal, that is, for any α ∈ D, there exists (α , β , ε ) ∈ D such that h(α , β , ε ) ≥ α.

So, the net y κ defined by y κ := z h(κ) , for all κ ∈ D , is a subnet of z α . In particular, it converges to ξ. Moreover, d(b, y κ ) + ξ(y κ ) converges to zero.

Let κ := (α, β, ε) and κ := (α , β , ε ) be elements of D , satisfying κ ≤ κ . So, α ≥ β, which implies that |d(z α , z α ) -d(b, z α ) -ξ(z α )| < ε.

Hence, for κ and κ large enough, with κ < κ ,

d(y κ , y κ ) -d(b, y κ ) < ξ(y κ ) + ε < -d(b, y κ ) + 2ε,
which proves that y κ is an almost-geodesic.

The detour cost satisfies the triangle inequality and is non-negative. By symmetrising the detour cost, we obtain a metric on the set of Busemann points:

δ(ξ, η) := H(ξ, η) + H(η, ξ),
for all Busemann points ξ and η.

We call δ the detour metric. It is possibly infinite valued, so it actually an extended metric. One may partition the set of Busemann points into disjoint subsets in such a way that δ(ξ, η) is finite if and only if ξ and η lie in the same subset. We call these subsets the parts of the horofunction boundary.

The following expression for the detour cost will prove useful. See [START_REF] Walsh | Gauge-reversing maps on cones, and Hilbert and Thompson isometries[END_REF]Prop. 4.5].

Proposition 2.10. Let ξ be a Busemann point, and η a horofunction of a metric space (X, d). Then,

H(ξ, η) = sup x∈X η(x) -ξ(x) = inf λ ∈ R | η(•) ≤ ξ(•) + λ .

The Busemann points of a normed space

In this section, we determine the Busemann points of an arbitrary normed space.

Let K be a convex subset of a locally-convex topological vector space E. A function f defined on K is said to be affine if

f (1 -λ)x + λy = (1 -λ)f (x) + λf (y),
for all x, y ∈ K and λ ∈ [0, 1]. We denote by A(K, E) the set of affine functions on K that are the restrictions of continuous affine functions on the whole of E. The following is [2, Cor. I.1.4] Lemma 3.1. If K is a compact convex subset of E and a : K → (-∞, ∞] is a lower semicontinuous affine function, then there is a non-decreasing net in A(K, E) converging pointwise to a.

The following lemma extends Dini's theorem. Lemma 3.2. Let g α be an almost non-increasing net of functions on a Hausdorff space D. Then, g α converges pointwise to a function g. If the g α are upper-semicontinuous, then so is the limit. If furthermore D is compact, then sup g α converges to sup g.

Proof. Let g α be an almost non-increasing net of functions on D, and choose x ∈ D. It is clear that for each ε > 0, we have lim inf α g α (x) ≥ lim sup α g α (x) -ε, from which it follows that g α (x) converges. Denote by g the pointwise limit of g α . Let x β be a net in D converging to a point x ∈ D, and let ε > 0. Assume that each g α is upper semicontinuous, and so g α (x) ≥ lim sup β g α (x β ). That g α is almost non-increasing implies that g ≤ g α + ε for α large enough. So, choose α large enough that this holds and g α (x) ≤ g(x) + ε. Putting all this together, we get g(x) ≥ lim sup β g(x β ) -2ε, and we conclude that g is upper semicontinuous. Now assume that D is compact. So, for each α, since g α is upper semicontinuous, it attains its supremum at some point x α , and furthermore the net x α has a cluster point x in D. By passing to a subnet if necessary, we may assume that g α (x α ) converges to a limit l, and that x α converges to x.

Let ε > 0. For α large enough, g α (x α ) ≤ g α (x α ) + ε for all α ≥ α. Taking the limit supremum in α , using the upper semicontinuity of g α , and then taking the limit in α we get that l ≤ g(x) + ε ≤ sup g + ε, and hence that l ≤ sup g, since ε was chosen arbitrarily. The opposite inequality comes from the fact that, in general, the limit of a supremum is greater than or equal to the supremum of the limit.

Let (X, || • ||) be a normed space. We denote by B the unit ball of X, and by B • the dual ball. The topological dual space of X is denoted by X * , and we take the weak * topology on this space. The dual ball is compact in the weak * topology, by the Banach-Alaoglu theorem.

Recall that the Legendre-Fenchel transform of a function f on X is defined to be

f * (y) := sup x∈X y, x -f (x) , for all y ∈ X * .
Since it is a supremum of weak* continuous affine functions, f * is weak* lower semicontinuous and convex. One may also define the transform of a function g on X * as follows:

g * (x) := sup y∈X * y, x -g(x) , for all x ∈ X.
These maps are inverses of one another in the following sense. A function taking values in (-∞, ∞] is said to be proper if it not identically ∞. Recall that a lower semicontinuous convex function is automatically weakly lower semicontinuous. Denoting by Γ(X) the proper lower semicontinuous convex functions on X, and by Γ * (X * ) the proper weak* lower semicontinuous convex functions on X * , we have

f * * = f for f ∈ Γ(X)
and

g * * = g for g ∈ Γ * (X * ).
We use the notation f | G to denote the restriction of a function f to a set G. Also, we denote by X B (∞) the set of Busemann points of a metric space X.

Theorem 3.3. Let (X, || • ||) be a normed space. A function on X is in X B (∞) ∪ X if
and only if it is the Legendre-Fenchel transform of a function that is affine on the dual ball, infinite outside the dual ball, and weak* lower semi-continuous, and has infimum 0.

Proof. The Legendre-Fenchel transform of any function is automatically weak* lower semi-continuous. Every Busemann point is 1-Lipschitz, and so its transform takes the value ∞ outside the dual ball. Since each Busemann point takes the value 0 at the origin, the transform has infimum 0. That the transform of a Busemann point must be affine on the dual ball was proved in [START_REF] Walsh | The horofunction boundary of finite-dimensional normed spaces[END_REF]Lemma 3.1]; the theorem is stated there for finite dimensional spaces, but the proof works in infinite dimension as well. Now let f be a real-valued function on X such that its transform f * has the properties stated. By Lemma 3.1, there exists a non-decreasing net g α of elements of A(B • , X * ) that converges pointwise to f * . For each α, we may write

g α = •, z α | B • + c α , where z α ∈ X and c α ∈ R.
Let m α := inf g α , for each α. So, m α is a non-decreasing net of real numbers, and by Lemma 3.2 it converges to inf f * = 0. It is not too hard to calculate that, for each α, the transform of [START_REF] Walsh | The horofunction boundary of finite-dimensional normed spaces[END_REF]. The Legendre-Fenchel transform is order-reversing, and so the net (g α ) * is nonincreasing. So, by the observation after the Definition 2.2, (g α ) * + m α is almost nonincreasing. But

ϕ zα (•) := ||z α -• || -||z α || is ϕ * zα = g α -m α ; see
(g α ) * + m α = (g α -m α ) * = ϕ zα , for all α.
Therefore, by Proposition 2.4, z α is an almost geodesic in (X, || • ||).

Let x be a point in X. We have

ϕ zα (x) = m α + sup y∈B • y, x -g α (y) , for all α.
Since g α is non-decreasing, the net of functions •, x -g α (•) is non-increasing. So, by Lemma 3.2, its supremum over the dual ball B • converges to the supremum of the pointwise limit •, x -f * (•). We deduce that ϕ zα converges pointwise to f . We have thus proved that f is either a Busemann point or a point in the horofunction compactification corresponding to an element of X.

We now determine the detour metric on the boundary of a normed space.

Theorem 3.4. Let ξ 1 and ξ 2 be Busemann points of a normed space, having Legendre-Fenchel transforms g 1 and g 2 , respectively. Then, the distance between them in the detour metric is

δ(ξ 1 , ξ 2 ) = sup y∈B • g 1 (y) -g 2 (y) + sup y∈B • g 2 (y) -g 1 (y) .
Proof. By the properties of the Legendre-Fenchel transform, we have, for any λ ∈ R, that ξ 2 ≤ ξ 1 + λ if and only if g 2 ≥ g 1 -λ. So, applying Proposition 2.10, we get

H(ξ 1 , ξ 2 ) = inf λ ∈ R | g 2 ≥ g 1 -λ = sup y∈B • g 1 (y) -g 2 (y) .
The result is now obtained upon symmetrising.

Corollary 3.5. Two Busemann points of a normed space are in the same part if and only if their respective Legendre-Fenchel transforms g 1 and g 2 satisfy

g 1 -c ≤ g 2 ≤ g 1 + c, for some c ∈ R. ( 4
)
Corollary 3.6. A function ξ is a singleton Busemann point of a normed space if and only if it is an extreme point of the dual ball.

Proof. If ξ 1 is an extreme point of the dual ball, then its transform g 1 takes the value zero at ξ 1 and infinity everywhere else. Let ξ 2 be another Busemann point in the same part, which implies that its transform g 2 satisfies (4). So, g 2 is finite at ξ 1 and infinite everywhere else, and since, by Theorem 3.3, it has infimum zero, we get that g 2 = g 1 . Hence ξ 2 is identical to ξ 1 . Now let ξ 1 be a Busemann point that is not an extreme point of the dual ball, and let g 1 be its transform. Since g 1 is affine on the dual ball, the set on which it is finite is an extreme set of this ball, and therefore must contain at least two points, for otherwise ξ 1 would be an extreme point. Choose an element x of the normed space such that •, x separates these two points, that is, does not take the same value at the two points. The function

g 2 := g 1 + •, x -inf B • (g 1 + •, x
) satisfies all the conditions of Theorem 3.3, and so its transform ξ 2 is a Busemann point. Moreover, g 1 and g 2 satisfy (4), which implies that ξ 2 is in the same part as ξ 1 . But, by construction, g 2 differs from g 1 , and so ξ 2 differs from ξ 1 .

The Masur-Ulam Theorem

The techniques developed so far allow us to write a short proof of the Masur-Ulam theorem.

Recall that, according to Corollary 3.6, the singleton Busemann points of a normed space are exactly the extreme points of the dual ball. Recall also that any surjective isometry between metric spaces can be extended to a homeomorphism between their horofunction boundaries, which maps singletons to singletons. Theorem 4.1 (Masur-Ulam). Let ϕ : X → Y be a surjective isometry between two normed spaces. Then, ϕ is affine.

Proof. It will suffice to assume that ϕ maps the origin of X to the origin of Y , and show that it is linear.

Let f be an extreme point of the dual ball of Y . So, f is a singleton of the horofunction boundary of Y . Therefore, f • ϕ is a singleton of the horofunction boundary of X, and hence an extreme point of the dual ball of X, and hence linear. So,

f ϕ(αx + βy) = αf (ϕ(x)) + βf (ϕ(y)) = f (αϕ(x) + βϕ(y)),
for all α, β ∈ R and x, y ∈ X.

Since this is true for every extreme point f of the dual ball of Y , we have ϕ(αx + βy) = αϕ(x) + βϕ(y).

The horofunction boundary of

(C(K), || • || ∞ )
In this section we look in more detail at the space C(K) with the supremum norm, where K is an arbitrary compact Hausdorff space. Here we can describe explicitly the Busemann points.

Theorem 5.1. The Busemann points of (C(K), ||•|| ∞ ) are the functions of the following form

Φ(g) := sup x∈K -u(x) -g(x) ∨ sup x∈K -v(x) + g(x) , for all g ∈ C(K), (5)
where u and v are two lower-semicontinuous functions from

K → [0, ∞], such that inf u ∧ inf v = 0, and such that u(x) ∨ v(x) = ∞ for all x ∈ K.
The proof will use the characterisation in the previous section of the Legendre-Fenchel transforms of the Busemann points of a normed space. Recall that these were shown to be the functions that are affine on the dual ball, infinite outside the dual ball, weak* lower semi-continuous, and have infimum 0. We will identify all such functions on the dual space of C(K).

Recall that the dual space of C(K) is ca r (K), the set of regular signed Borel measures on K of bounded variation. Any element µ of ca r (K) can be written µ = µ + -µ -, where µ -and µ + are non-negative measures. This is called the Jordan decomposition. The dual norm is the total variation norm, which satisfies ||µ|| = ||µ -|| + ||µ + ||. Proposition 5.2. Consider a function from Θ : ca r (K) → R + that is not the restriction to the dual ball of a continuous affine function. Then, Θ is affine on the dual ball, infinite outside the dual ball, weak* lower semi-continuous, and has infimum 0 if and only if it can be written

Θ(µ) = Ξ(µ) :=    +∞, ||µ|| = 1; u dµ -+ v dµ + , ||µ|| = 1, (6)
where u and v are as in the statement of Theorem 5.1.

The proof if this proposition will require several lemmas. Proof. Let µ α be a net in ca r (K) converging in the weak* topology to µ ∈ ca r (K). We must show that lim inf α Ξ(µ α ) ≥ Ξ(µ). By taking a subnet if necessary, we may suppose that Ξ(µ α ) converges to a limit, which we assume to be finite. This implies that ||µ α || = 1, eventually. Since the dual unit ball is compact, we may, by taking a further subnet if necessary, assume that µ + α and µ - α converge, respectively, to non-negative measures ν and ν . These measures satisfy µ = µ + -µ -= ν -ν , and, so, from the minimality property of the Jordan decomposition, ν ≥ µ + and ν ≥ µ -.

Since u and v are lower-semicontinuous, we get from the Portmanteau theorem that

lim inf α Ξ(µ α ) ≥ lim inf α u dµ - α + lim inf α v dµ + α ≥ u dν + v dν. (7)
Consider the case where μ := ν -µ + = ν -µ -is non-zero. Since u + v is identically infinity, either u dμ or v dμ must equal infinity. This implies that the right-hand-side of ( 7) is equal to infinity.

On the other hand, if ν = µ + and ν = µ -, then

||µ|| = ||µ + || + ||µ -|| = ||ν|| + ||ν || = lim α ||µ α || = 1.
So, in this case, the right-hand-side of ( 7) is equal to Ξ(µ).

Lemma 5.4. Let µ 1 and µ 2 be in the unit ball of ca r (K).

Let µ := (1 -λ)µ 1 + λµ 2 , for some λ ∈ (0, 1). If ||µ|| = 1, then ||µ 1 || = ||µ 2 || = 1, and µ + = (1 -λ)µ + 1 + λµ + 2 and µ -= (1 -λ)µ - 1 + λµ - 2 .
Proof. Observe that the functions µ → ||µ + || and µ → ||µ -|| are both convex, and hence

||µ + || ≤ (1 -λ)||µ + 1 || + λ||µ + 2 || and (8) ||µ -|| ≤ (1 -λ)||µ - 1 || + λ||µ - 2 ||. (9)
Moreover, the sum of these two functions is µ → ||µ||. Using that ||µ 1 || ≤ 1 and ||µ 2 || ≤ 1, and ||µ|| = 1, we deduce that inequalities ( 8) and ( 9) are actually equalities.

Since µ → µ + is also convex, we have

µ + ≤ (1 -λ)µ + 1 + λµ + 2 .
Combining this with the equalities just established, we get that µ + = (1 -λ)µ + 1 + λµ + 2 , since the norm is additive on non-negative measures. The equation involving µ -is proved similarly.

Lemma 5.5. The function Ξ is affine on the unit ball of ca r (K).

Proof. Let µ, µ 1 , and µ 2 be in the unit ball of ca r (K), such that µ = (1 -λ)µ 1 + λµ 2 , for some λ ∈ (0, 1). We wish to show that Ξ(µ) = (1 -λ)Ξ(µ 1 ) + λΞ(µ 2 ).

Consider the case when ||µ|| = 1. By Lemma 5.4,

||µ 1 || = ||µ 2 || = 1, and µ + = (1 -λ)µ + 1 + λµ + 2 and µ -= (1 -λ)µ - 1 + λµ - 2 .
We deduce that the second case in the definition of Ξ is the relevant one, for each of µ, µ 1 , and µ 2 , and furthermore that the affine relation holds. Now consider the case where ||µ|| < 1. So, Ξ(µ) = ∞. To prove the affine relation, we must show that either Ξ(µ 1 ) or Ξ(µ 2 ) is infinite.

Assume for the sake of contradiction that both Ξ(µ 1 ) and Ξ(µ 2 ) are finite. Denote by U and V the subsets of K where, respectively, u and v are finite. So, U and V are disjoint. From the definition of Ξ, we see that ||µ 1 || = ||µ 2 || = 1, that µ + 1 and µ + 2 are concentrated on V , and that µ -

1 and µ - 2 are concentrated on U . It follows that ||µ|| = (1 -λ)(||µ + 1 || + ||µ - 1 ||) + λ(||µ + 2 || + ||µ - 2 ||) = 1
, which contradicts our assumption. The following result is due to Choquet; see [2, Theorem I.2.6].

Theorem 5.6. If f is a real-valued affine function of the first Baire class on a compact convex set K in a locally convex Hausdorff space, then f is bounded and f (x) = f dµ, where µ is any probability measure on K and x is the barycenter of µ.

We will need a version of Lebesgue's monotone convergence theorem for nets of functions. The following was proved in [5, Proposition 2.13].

Lemma 5.7. Let X be a locally compact and σ-compact Hausdorff space, and let λ be a positive Borel measure that is complete and regular and satisfies λ(K) < ∞ for all compact sets K ⊂ X.

Let I be a directed set, and let f i : X → [0, ∞], i ∈ I, be a family of lower semicontinuous functions that is monotone non-decreasing. Set f (x) := sup i∈I f i (x) for all x ∈ X. Then,

X f dλ = sup i∈I X f i dλ.
Lemma 5.8. If a function Ξ : ca r (K) → R + is affine on the dual ball, infinite outside the dual ball, weak* lower semi-continuous, and has infimum 0, then it either can be written in the form [START_REF] Beer | Topologies on closed and closed convex sets[END_REF] or is the restriction to the dual ball of a continuous affine function on the dual space.

Proof. Denote by δ x the measure consisting of an atom of mass one at a point x. Define the functions

v : K → [0, ∞], v(x) := Ξ(δ x ), and
u : K → [0, ∞], u(x) := Ξ(-δ x ).
These two functions are non-negative because inf Ξ = 0. Moreover, the dual ball is weak* compact, and so, as a lower semicontinuous affine function, Ξ attains its infimum over it at an extreme point. Recall that, in the present case, the extreme points are exactly the positive and negative Dirac masses; denote the set of these by

∂ e := ∂ + e ∪ ∂ - e , where ∂ + e := {δ x | x ∈ K} and ∂ - e := {-δ x | x ∈ K}. Thus, inf u ∧ inf v = 0.
Also observe that u and v are lower semicontinuous. Consider the case where u(x) and v(x) are both finite for some x ∈ K. This implies that Ξ(0) is finite since Ξ is affine. It follows from this that Ξ is finite on the whole of the dual ball. Using the fact that the dual ball is balanced, that is, closed under multiplication by scalars of absolute value less than or equal to 1, we can reflect about the origin to get that Ξ is upper semicontinuous. So, Ξ is continuous on the dual ball. It is hence the restriction of a continuous affine function on the whole dual space; see [2, Cor. I.1.9].

So, from now on, assume that u(x) ∨ v(x) = ∞, for all x ∈ K Since Ξ is affine on the dual ball and infinite outside it, the set where Ξ is finite is an extreme set of the dual ball. Note that, given any distinct points µ 1 and µ 2 in the dual ball such that ||µ 2 || < 1, there is a line segment in the dual ball having µ 1 as a endpoint and µ 2 as a point in the relative interior. It follows that if Ξ is finite at some point µ 2 with ||µ 2 || < 1, then Ξ is finite everywhere in the dual ball. But this contradicts what we have just assumed, and we conclude that Ξ(µ) takes the value +∞ if ||µ|| < 1.

Let µ be in the dual ball such that ||µ|| = 1. By Choquet theory, there is a probability measure µ on the dual ball that has barycenter µ and is pseudo-concentrated on the extreme points of the dual ball. In the present case, since the set of extreme points is closed, and hence measurable, µ is concentrated on the extreme points.

In fact, we have the following description of µ: for any measurable subset

U = {δ x | x ∈ U + ⊂ K} ∪ {-δ x | x ∈ U -⊂ K} of ∂ e , we have µ[U ] := µ + [U + ] + µ -[U -].
By Lemma 3.1, there is a non-decreasing net g α of continuous affine functions on the dual space converging pointwise to Ξ on the dual ball. Applying Theorem 5.6 and Lemma 5.7, we get

Ξ(µ) = lim α g α (µ) = lim α g α dµ = Ξ dµ = K Ξ(δ x ) dµ + + K Ξ(-δ x ) dµ - = K v(x) dµ + + K u(x) dµ -.
Finally, it's not hard to show that Ξ(µ) takes the value +∞ if ||µ|| < 1.

Proof of Proposition 5.2. Any function Ξ of the given form is clearly infinite outside the dual ball and has infimum zero. The rest was proved in Lemmas 5.3, 5.5, and 5.8.

Lemma 5.9. The function Ξ is the Legendre-Fenchel transform of the function Φ in Theorem 5.1.

Proof. Fix g ∈ C(K), and let Ψ : ca r (K) → R + be defined by Ψ(µ) := µ, g -Ξ(µ). By Lemmas 5.3 and 5.5, Ψ is upper-semicontinuous and affine on the unit ball of ca r (K). Outside the unit ball, Ψ takes the value -∞. So, the supremum of Ψ is attained at an extreme point of the unit ball. The set of these extreme points is

{δ x | x ∈ K} ∪ {-δ x | x ∈ K}. For all x ∈ K, we have that Ψ(δ x ) = g(x) -v(x) and Ψ(-δ x ) = -g(x) -u(x).
It follows that the Legendre-Fenchel transform of Ξ is the function Φ given in [START_REF] Baranov | Majorization in de Branges spaces. III. Division by Blaschke products[END_REF].

Since Ξ is a lower-semicontinuous proper convex function, it is equal to the transform of its transform.

We can now prove Theorem 5.1.

Proof of Theorem 5.1. We combine Theorem 3.3, Proposition 5.2, and Lemma 5.9.

The horofunction boundary of the reverse-Funk geometry

Although they are not strictly speaking metric spaces, the reverse-Funk and Funk geometries retain enough of the properties of metric spaces for the definition of the horofunction boundary, and of Busemann points, to make sense. In this and the following section, we study the boundary of these two geometries.

Recall that the indicator function I E of a set E is defined to take the value 1 on E and the value 0 everywhere else. Lemma 6.1. Let D be a compact convex subset of a locally-convex Hausdorff space E, and let f 1 and f 2 be upper-semicontinuous real-valued non-negative affine functions on D.

If sup D f 1 /g ≤ sup D f 2 /g for each continuous real-valued affine function g on E that is positive on D, then f 1 ≤ f 2 on D.
Proof. Let y be an extreme point of D. The function 1/ I {y} , which takes the value 1 at y and the value ∞ everywhere else, is a weak*-lower-semicontinuous affine function on D. Therefore, there exists a non-decreasing net g α of continuous real-valued affine functions that are positive on D such that g α converges pointwise on D to 1/ I {y} . So, both f 1 /g α and f 2 /g α are non-increasing nets of real-valued upper-semicontinuous functions on D converging pointwise, respectively, to f 1 I {y} and f 2 I {y} . By Lemma 3.2, sup D f 1 /g α and sup D f 2 /g α converge respectively to f 1 (y) and f 2 (y). We conclude that f 1 (y) ≤ f 2 (y). This is true for any extreme point of D, and the conclusion follows upon applying Choquet theory.

The next lemma is similar to the previous one. Proof. For each extreme point y of D, the indicator function I {y} is a weak*-uppersemicontinuous affine function on D. Therefore, there exists a non-increasing net g α of continuous real-valued positive affine functions on D that converges pointwise to I {y} . So, both g α /f 1 and g α /f 2 are non-increasing nets of functions converging pointwise, respectively, to I {y} /f 1 and I {y} /f 2 . By Lemma 3.2, sup D g α /f 1 and sup D g α /f 2 converge respectively to 1/f 1 (y) and 1/f 2 (y). We conclude that f 1 (y) ≥ f 2 (y). This is true for any extreme point of D, and the conclusion follows upon applying Choquet theory. Theorem 6.3. The Busemann points of the reverse-Funk geometry on C are the functions of the following form:

ξ(x) := log sup y∈C * \{0} g(y) y, x , for all x ∈ C, ( 10
)
where g is a weak*-upper-semicontinuous non-negative linear functional on C * , normalised so that sup y∈C * \{0} g(y)/ y, b = 1.

Proof. We consider the cross-section D := {y ∈ C * | y, b = 1}, which is compact. Each linear functional on C * corresponds to an affine function on D.

Let ξ be of the above form. Take a net g α of elements of C that, when viewed as a net of continuous affine functions on D, is non-increasing and converges pointwise to g. Fix x ∈ C. So, the function y → g α (y)/ y, x defined on D is non-increasing and converges pointwise to g(y)/ y, x . Therefore, by Lemma 3.2, the net d R (x, g α ) := log sup y∈D g α (y) y, x converges to ξ(x). In particular, d R (b, g α ) converges to zero. It follows that g α converges to ξ in the reverse-Funk horofunction boundary. Moreover, the monotonicity of the convergence implies that d R (•, g α )-d R (b, g α ) is an almost non-increasing net of functions; see the observation after Definition 2.2. So, by Proposition 2.4, g α is an almost-geodesic and ξ is a Busemann point. Now let g α be an almost-geodesic net in C converging to a Busemann point ξ. So, d R (•, g α ) -d R (b, g α ) is an almost non-increasing net of functions converging to ξ. By scaling g α if necessary, we may assume that d R (b, g α ) = 0, for all α.

So, for any ε > 0, there exists an index A such that sup y∈D g α (y) y, x ≤ e ε sup y∈D g α (y) y, x , for all x ∈ C, whenever α and α satisfy A ≤ α ≤ α . But this implies by Lemma 6.1 that g α ≤ e ε g α on D, whenever A ≤ α ≤ α . We conclude that log g α | D is an almost non-increasing net.

Applying Lemma 3.2 and exponentiating, we get that g α converges pointwise on D to an upper semicontinuous function g, which is necessarily affine and non-negative. We extend g to a linear functional on C * using homogeneity.

By applying Lemma 3.2 to the function log g α (•) -log •, x on D, we get that ξ, which is the pointwise limit of d R (•, g α ), has the form given in the statement of the theorem.

The normalisation can be verified by evaluating at b. Theorem 6.4. Let ξ 1 and ξ 2 be Busemann points of the reverse-Funk geometry, corresponding via [START_REF] Gromov | Hyperbolic manifolds, groups and actions[END_REF] to linear functionals g 1 and g 2 , respectively, with the properties specified in Lemma 6.3. Then, the distance between them in the detour metric is

δ(ξ 1 , ξ 2 ) = log sup y∈C * g 1 (y) g 2 (y) + log sup y∈C * g 2 (y) g 1 (y) .
(The supremum is always taken only over those points where the ratio is well-defined).

Proof.

Let D := {y ∈ C * | y, b = 1},
H(ξ 1 , ξ 2 ) = inf λ ∈ R | ξ 2 (•) ≤ ξ 1 (•) + λ = log sup y∈D g 2 (y) g 1 (y) .
The result is now obtained upon symmetrising.

Corollary 6.5. The two reverse-Funk Busemann points ξ 1 and ξ 2 are in the same part if and only if g 2 /λ ≤ g 1 ≤ λg 2 , for some λ > 0.

Questions 6.6. Is it possible for reverse-Funk geometries to have non-Busemann horofunctions? This is not the case in finite dimension [START_REF] Walsh | The horofunction boundary of the Hilbert geometry[END_REF], and we will see in section 9 that it is not the case either for the positive cone C + [0, 1].

The linear functional g in the statement of the theorem is always bounded on D and therefore continuous in the norm topology of the dual. Do singleton Busemann points correspond exactly the extreme rays of the bidual cone?

We have some partial results concerning the singletons Busemann points of this geometry. Proposition 6.7. Let ξ be a Busemann point of the reverse-Funk geometry, and let g be as in [START_REF] Gromov | Hyperbolic manifolds, groups and actions[END_REF]. If g is in an extremal ray of the bidual cone C * * , then ξ is a singleton Busemann point.

Proof. Let ξ 1 and ξ 2 be Busemann points in the same part. By Theorem 6.3, we may write both of these points in the form [START_REF] Gromov | Hyperbolic manifolds, groups and actions[END_REF], with g 1 and g 2 , respectively, substituted in for g. Both g 1 and g 2 are bounded on the cross-section D of the dual cone, and therefore continuous in the norm topology of the dual. Thus, they are both elements of the bidual. By Corollary 6.5, there exists λ > 0 such that g 2 /λ ≤ g 1 ≤ λg 2 . Define f = g 1 + g 2 /λ and h = g 1 -g 2 /λ. Both f and h are linear functionals on the dual cone that are continuous in the norm topology of the dual space. Moreover, they are non-negative. We conclude that f and h are in the bidual cone. But we have g 1 = f /2 + h/2, which shows that g 1 is not in an extremal ray of the bidual cone.

Let U be the cone of non-negative finite weak * -upper-semicontinuous linear functionals on the dual cone C * . Proposition 6.8. Let ξ be a Busemann point of the reverse-Funk geometry, and let g be as in [START_REF] Gromov | Hyperbolic manifolds, groups and actions[END_REF]. If ξ is a singleton, then g is in an extremal ray of the cone U .

Proof. Let g = g 1 + g 2 , with g 1 and g 2 in U , and write g := g 1 + 2g 2 . By normalising g , the conditions of Theorem 6.3 are met, so we obtain a Busemann point ξ . Moreover, g /2 ≤ g ≤ g , and so according to Theorem 6.4, ξ and ξ lie in the same part of the boundary. Therefore, ξ = ξ , which implies that g is a multiple of g, which further implies that g 1 is a multiple of g 2 .

The horofunction boundary of the Funk geometry

The proof of the following theorem parallels that of the corresponding result for the reverse-Funk geometry.

Theorem 7.1. The Busemann points of the Funk geometry on C are the functions of the following form:

ξ(x) := log sup y∈C * \{0} y, x f (y) , for all x ∈ C. ( 11
)
where f is a weak*-lower-semicontinuous non-negative linear function on the dual cone C * , normalised so that sup y∈C * \{0} y, b /f (y) = 1.

Proof. We consider the cross-section D := {y ∈ C * | y, b = 1}, which is compact. Each linear functional on C * corresponds to an affine function on D.

Let ξ be of the above form. Take a net f α of elements of C that, when viewed as a net of continuous affine functions on D, is non-decreasing and converges pointwise to f . Fix x ∈ C. So, the function y → y, x /f α (y) is non-increasing and converges pointwise to y, x /f (y). Therefore, by Lemma 3.2, the net

d F (x, f α ) := log sup y∈D y, x f α (y)
converges to ξ(x). In particular, d F (b, f α ) converges to zero. It follows that f α converges to ξ in the Funk horofunction boundary. Moreover, the monotonicity of the convergence implies that d F (•, f α ) -d F (b, f α ) is an almost non-increasing net of functions; see the observation after Definition 2.2). So, by Proposition 2.4, f α is an almost-geodesic and ξ is a Busemann point. Now let f α be an almost-geodesic net in C converging to a Busemann point ξ. So, d F (•, f α ) -d F (b, f α ) is an almost non-increasing net of functions converging to ξ. By scaling f α if necessary, we may assume that d F (b, f α ) = 0, for all α.

So, for any ε > 0, there exists an index A such that sup y∈D y, x f α (y) ≤ e ε sup y∈D y, x f α (y) , for all x ∈ C, whenever α and α satisfy A ≤ α ≤ α . But this implies by Lemma 6.2 that e ε f α ≥ f α on D, whenever A ≤ α ≤ α . We conclude that log f α | D is an almost non-decreasing net.

Applying Lemma 3.2 and exponentiating, we get that f α converges pointwise on D to a lower-semicontinuous function f , which is necessarily affine and non-negative. We extend f to a linear functional on C * using homogeneity.

By applying Lemma 3.2 to the function log •, x -log f α (•) on D, we get that ξ, which is the pointwise limit of d F (•, f α ), has the form given in the statement of the theorem.

The normalisation can be verified by evaluating at b.

The proofs of the following theorem and corollary are similar to those of the equivalent results in the case of the reverse-Funk geometry.

Theorem 7.2. Let ξ 1 and ξ 2 be Busemann points of the reverse-Funk geometry, corresponding via [START_REF] Lemmens | Isometries of infinite dimensional Hilbert geometries[END_REF] to linear functionals f 1 and f 2 , respectively, with the properties specified in Lemma 7.1. Then, the distance between them in the detour metric is

δ(ξ 1 , ξ 2 ) = log sup y∈C * f 1 (y) f 2 (y) + log sup y∈C * f 2 (y) f 1 (y) .
(The supremum is always taken only over those points where the ratio is well-defined). Proof. The proof is the same as that of Corollary 3.6, when one considers the cross-section D instead of the dual ball.

The horofunction boundary of the Hilbert geometry

In this section, we relate the boundary of the Hilbert geometry to those of the reverse-Funk and Funk geometries.

We denote by P (C) the projective space of the cone C, and by [h] the projective class of an element h of C. Recall that we may regard the elements of C as positive continuous linear functionals of C * . Proposition 8.1. Let z α be a net in C. Then, z α is an almost-geodesic in the Hilbert geometry if and only if it is an almost-geodesic in both the Funk and reverse-Funk geometries.

Proof. For α and α satisfying α < α , define

R(α, α ) := d R (b, z α ) + d R (z α , z α ) -d R (b, z α ), F (α, α ) := d F (b, z α ) + d F (z α , z α ) -d F (b, z α ), and
H(α, α ) := d H (b, z α ) + d H (z α , z α ) -d H (b, z α ).
Clearly H = R + F . Also, by the triangle inequality, R, F , and H are all non-negative.

For any α and α with α < α , and any ε > 0, we have that H(α, α ) < ε implies R(α, α ) < ε and F (α, α ) < ε. Conversely, we have that R(α, α ) < ε/2 and F (α, α ) < ε/2 implies H(α, α ) < ε. The conclusion follows easily.

We define the following compatibility relation between reverse-Funk and Funk horofunctions. We write ξ R ∼ ξ F , when there exists a net in C that is an almost-geodesic in both the reverse-Funk and the Funk geometry, and converges to ξ R in the former and to ξ F in the latter.

Theorem 8.2. The Busemann points of the Hilbert geometry are the functions of the form ξ H := ξ R +ξ F , where ξ R and ξ F are, respectively, reverse-Funk and Funk Busemann points, satisfying ξ R ∼ ξ F . Each Hilbert Busemann point may we written in a unique way in this form.

Proof. Let ξ R and ξ F be as in the statement. So, there exists a net z α in P (C) that is an almost-geodesic in both reverse-Funk geometry and in the Funk geometry, and converges to ξ R in the former and to ξ F in the latter. Applying Proposition 8.1, we get that z α is an almost-geodesic in the Hilbert geometry, and it must necessarily converge to ξ R + ξ F .

To prove the converse, let ξ H be a Busemann point of the Hilbert geometry, and let z α be an almost-geodesic net converging to it. By Proposition 8.1, z α is an almost-geodesic in both the Funk and reverse-Funk geometries, and so converges to a Busemann point ξ F in the former and to a Busemann point ξ R in the latter. So, ξ R ∼ ξ F , and we also have that ξ

H = ξ R + ξ F .
To prove the last part, suppose that ξ H = ξ R + ξ F = ξ R + ξ F , where ξ R and ξ R are reverse-Funk Busemann points and ξ F and ξ F are Funk Busemann points, with ξ R ∼ ξ F and ξ R ∼ ξ F . So, there exist nets z α and z α in P (C) that are almost geodesic in both the reverse-Funk and Funk geometries, and converge in the former to ξ R and ξ R , respectively, and in the latter to ξ F and ξ F , respectively. By Proposition 8.1, both z α and z α are almost geodesic in the Hilbert geometry, in which they converge to ξ H . By Proposition 2.5, we can find Hilbert almost-geodesics x β , x β , and y β such that x β is a subnet of both z α and y β , and x β is a subnet of both z α and y β . Each of these three nets is an almost-geodesic in the Funk geometry, by Proposition 8.1, and so must have a limit in this geometry. Recalling that a subnet of a convergent net has the same limit, it is not too hard to see that the three nets above and the nets z α and z α must all have the same limit in the Funk geometry. The same reasoning also holds for the reverse-Funk geometry. It follows that ξ R = ξ R and ξ F = ξ F . Our next goal is to make explicit the meaning of the relation ∼. Suppose we are given two non-negative linear functionals f and g on C * with the following properties. We assume that g is upper semicontinuous and has supremum 1 on the cross-section D, whereas f is lower semicontinuous and has infimum 1 on the same set. We assume further that g takes the value zero everywhere that f is finite.

Denote by C the set

C := (h, h ) ∈ C × C | [h] = [h ] and g < h ≤ h < f .
We define a relation on C in the following way: we say that (h

1 , h 1 ) (h 2 , h 2 ) if h 2 ≤ h 1 and h 1 ≤ h 2 .
It is clear that this ordering is reflexive, transitive, and antisymmetric.

epi f := (x, λ) ∈ D × R | f (x) ≤ λ and hyp g := (x, λ) ∈ D × R | g(x) ≥ λ ≥ 0 .
Both of these sets are closed and convex, and hyp g is compact.

Let x ∈ D, and let λ < f (x). Let K be the convex hull of the union of {(x, λ)} and hyp g. It is not hard to check that K is compact and disjoint from epi f . Therefore, by the Hahn-Banach separation theorem, there is a closed hyperplane H in E × R that strongly separates epi f and K. Note that the strong separation implies that H can not be of the form H × R, where H is a hyperplane of E. It follows that H is the graph of a continuous affine function h : E → R, satisfying g < h < f and h(x) > λ.

We can extend h to the whole of the dual space in a unique way by requiring homogeneity. Since h is strictly positive on D, this gives us an element of C, which we denote again by h. So (h, h) ∈ C. We have established that f (x) ≥ sup{h (x) | (h, h ) ∈ C} when x ∈ D. The same is true for any x ∈ C * , by homogeneity. The opposite inequality is trivial.

The second part is similar, but we must be careful because the epigraph of f is not necessarily compact. So, this time, we choose λ arbitrarily so that λ > g(x), and separate hyp g from the convex hull of the following three sets,

epi f ∩ {(y, β) ∈ D × R | 0 ≤ β ≤ 2}, {(y, β) ∈ D × R | β = 2}, and {(x, λ)}.
All three of these sets are compact, and, since none of them intersect hyp g, neither does the convex hull of their union.

In the same manner as before, we obtain an element h of C satisfying g < h < min(2, f ) on D, and h(x) < λ, and the rest of the proof is the same. Lemma 8.4. Let f be a lower-semicontinuous linear functional on C * , and let {h i } i be a finite collection of upper-semicontinuous linear functionals on C * satisfying h i < f on the cross-section D, for each i. Then there exists a h ∈ C such that max i h i < h < f on D.

Proof. The proof is similar to that of the previous lemma. We chose I ∈ (-∞, inf f ), and separate epi f from the convex hull of the union of the compact sets

hyp g i ∩ {(y, β) ∈ D × R | I ≤ β}, for all i, and {(y, β) ∈ D × R | β = I}.
We obtain h ∈ C satisfying I < h < f , and h(y) > g i (y) for all i and y ∈ D such that g i (y) ≥ I. The conclusion follows.

Lemma 8.5. The set C is a directed set under the ordering .

Proof. Let (h 1 , h 1 ) and (h 2 , h 2 ) be in C.

By Lemma 8.4, there is a continuous real-valued linear functional h satisfying

max(h 1 , h 2 ) < h < f.
Restrict attention to the compact cross-section D of C * , on which the linear functionals are affine functions. Since g is upper-semicontinuous, and h 1 and h 2 are continuous, the function min(h 1 , h 2 ) -g attains its minimum over D. This minimum is positive. Choose an ε ∈ (0, 1) strictly smaller than this minimum. Let λ ∈ (0, 1) be such that 0 < λh < ε. So, k := g + λh is a non-negative upper-semicontinuous linear functional. We have that max(g, λh 1 , λh 2 ) < k < min(h 1 , h 2 ). Also, since g takes the value zero everywhere that f is finite, we have k < λf .

We deduce using Lemma 8.4 that there exists a real-valued continuous linear functional l satisfying k < l < min(h 1 , h 2 , λf ). Hence g < l. Moreover,

max(h 1 , h 2 ) < l λ < f.
We have thus proved that (l, l/λ) is in C, and that

(h 1 , h 1 ) l, l λ and (h 2 , h 2 ) l, l λ .
Proposition 8.6. Let C be a cone giving rise to a complete Hilbert geometry. Let ξ R and ξ F be, respectively, a reverse-Funk Busemann point (of the form (10) and a Funk Busemann point (of the form [START_REF] Lemmens | Isometries of infinite dimensional Hilbert geometries[END_REF]. Then, ξ R ∼ ξ F if and only if, for each y ∈ C * \{0}, either g(y) = 0 or f (y) = ∞.

Proof. Assume that g and f satisfy the stated condition. The set C with the ordering defined using g and f is a directed set, by Lemma 8.5. Consider the net z α defined on the directed set C by z α := α, for all α ∈ C. Write (g α , f α ) := z α , for each α ∈ C. Observe that g α is non-increasing, and f α is non-decreasing.

Combining Lemmas 3.2 and 8.3 applied to a cross section D of C * , we get that the net f α converges to f . Fix x ∈ C. So, the net of functions y → y, x /f α (y) is non-increasing and converges pointwise to y, x /f (y). Therefore, by Lemma 3.2, the net

d F (x, f α ) = log sup y∈D y, x f α (y)
converges to ξ F (x). In particular, d F (b, f α ) converges to zero. It follows that f α converges to ξ F in the compactification of the Funk geometry. Moreover, the monotonicity of the convergence implies that d F (•, f α )-d F (b, f α ) is an almost non-increasing net of functions (see the observation after Definition 2.2). So, by Proposition 2.4, f α is an almost-geodesic.

Recall that convergence in the Funk geometry is a property of the projective class of the points rather that the points themselves. So, [f α ] converges in the Funk geometry to the Funk Busemann point ξ F .

The same method works to show that [g α ] converges in the reverse-Funk geometry to the reverse-Funk Busemann point ξ R . Recall, moreover, that [f α ] = [g α ], for all α.

To prove the converse, assume that ξ R and ξ F satisfy ξ R ∼ ξ F . So, there is a net z α in C that is an almost-geodesic in both the Funk and reverse-Funk geometries, and converges to ξ F in the former and to ξ R in the latter.

By using reasoning similar to that in second part of the proof of Theorem 6.3, we get that

z α / exp(d R (b, z α )) converges pointwise to g on C * . Similarly, z α exp(d F (b, z α )) converges pointwise to f . It follows that d H (b, z α ) := d R (b, z α ) + d F (b, z α ) converges to log(f (y)/g(y)
), for all y ∈ C * . But this net grows without bound according to Proposition 2.6, and so the latter function is identically infinity. We have shown that, at each point of C * \{0}, either g is zero, or f is infinite. Theorem 8.7. Let ξ H = ξ R + ξ F and ξ H := ξ R + ξ F be Busemann points of a Hilbert geometry, each written as the sum of a reverse-Funk Busemann point and a Funk Busemann point that are compatible with one another. Then, the distance between them in the detour metric is

δ H (ξ H , ξ H ) = δ R (ξ R , ξ R ) + δ H (ξ R , ξ R ),
where δ R and δ F denote, respectively, the detour metrics in the reverse-Funk and Funk geometries.

Proof. Let z α be a net in P (C) that is an almost-geodesic in the Hilbert geometry and converges in this geometry to ξ H . By Proposition 8.1, z α is also an almost-geodesic in both the reverse-Funk and Funk geometries. Moreover, it converges in the former geometry to ξ R and in the latter to ξ F . By Lemma 2.8, we have

H R (ξ R , ξ R ) = lim α d R (b, z α ) + ξ R (z α )
and

H F (ξ F , ξ F ) = lim α d F (b, z α ) + ξ F (z α ) ,
where H R and H F denote, respectively, the detour cost in the reverse-Funk and Funk geometry. Adding, and using Lemma 2.8 again, we get that

H R (ξ R , ξ R ) + H F (ξ F , ξ F ) = H H (ξ H , ξ H )
, where H H is the Hilbert detour cost. The result follows upon symmetrising.

Proposition 8.8. Assume the Hilbert geometry is complete. Let ξ R and ξ R be reverse-Funk Busemann points in the same part, and let ξ F and ξ F be Funk Busemann points in the same part.

If ξ R ∼ ξ F , then ξ R ∼ ξ F .
Proof. This follows from combining Proposition 8.6 with Theorems 6.4 and 7.2.

Corollary 8.9. A Busemann point ξ H = ξ R + ξ F of a complete Hilbert geometry, with ξ R ∼ ξ F , is a singleton if and only if ξ R and ξ F are singleton Busemann points of, respectively, the reverse-Funk and Funk geometries.

Proof. Assume that ξ R is not a singleton, that is, there exists another reverse-Funk Busemann point ξ R in the same part as it. By, Proposition 8.8, ξ R ∼ ξ F , and so, by Theorem 8.2, ξ R + ξ F is a Busemann point of the Hilbert geometry. Hence, ξ H is not a singleton. One may also prove in the same way that if ξ F is not a singleton, then neither is ξ H . Assume now that there exists a Busemann point ξ H = ξ R +ξ F , of the Hilbert geometry, with ξ R ∼ ξ F , that is distinct from ξ H but in the same part as it. So, either ξ R and ξ R are distinct, or ξ F and ξ F are. By Theorem 8.7, ξ R is in the same part as ξ R , and ξ F is in the same part as ξ F . This shows that either ξ R or ξ F is not a singleton. 9. The Hilbert geometry on the cone C + (K)

In this section, we study the positive cone C + (K), that is, the cone of positive continuous functions on a compact Hausdorff space K. We take the basepoint b to be the function that is identically equal to 1. The dual cone of C + (K) is the cone ca + r (K) of regular Borel measures on K. The cross section of the dual cone consisting of the probability measures on K is a Bauer simplex. The extreme points of this cross section are the Dirac masses. Every semicontinuous function on K can be extended to an semicontinuous linear functional on the dual cone: ḡ(µ) := K g dµ, for µ ∈ ca + r (K).

In the opposite direction, any semicontinuous linear functional on the dual cone gives rise to a semicontinuous function of K by being evaluated at the Dirac masses: g(x) = ḡ(δ x ), for x ∈ K. Recall that the hypograph of a function f , for all f ∈ C + (K), [START_REF] Lemmens | Isometries of polyhedral Hilbert geometries[END_REF] where g : K → [0, 1] is an upper-semicontinuous function with supremum 1. All these horofunctions are Busemann points.

: X → [-∞, ∞] is the set hyp f := {(x, α) ∈ X × R | α ≤ f (x)}. A net
Proof. Let ξ R be of the above form, extend g as above to get ḡ, a non-negative uppersemicontinuous linear functional on the dual cone. Let f ∈ C + (K), and denote by D the set of elements µ of the dual cone such that K f dµ = 1. So D is a cross-section of the dual cone. Observe that ḡ attains its supremum over D at an extreme point of D, and that the set of these extreme points is {δ x /f (x) | x ∈ K}, where δ x denotes the unit atomic mass at a point x. So,

ξ R (f ) = log sup µ∈D ḡ(µ) = log sup µ∈C * \{0} ḡ(µ) K f dµ .
Therefore, by Theorem 6.3, the function ξ R is a Busemann point of the reverse-Funk geometry of C + (K). Now let g α be a net in C + (K) converging to a horofunction. By scaling if necessary, we may assume that the supremum of each g α is 1. Consider the hypographs hyp(g α ) of these functions. This is a net of closed subsets of K × R. By the theorem of Mrowka (see [START_REF] Beer | Topologies on closed and closed convex sets[END_REF]Theorem 5.2.11,page 149]), this net has a subnet that converges in the Kuratowski-Painlevé topology. Therefore, g α has a subnet that converges in the hypograph topology to a proper upper-semicontinuous function g. We reuse the notation g α to denote this subnet.

Since g α takes values in [0, 1], so also does g. From [6, Theorem 5.3.6, page 160], we get that sup g α converges to sup g. Thus, sup g = 1.

Fix f ∈ C + (K). We have that g α /f converges in the hypograph topology to the proper upper semicontinuous function g/f . Applying [START_REF] Beer | Topologies on closed and closed convex sets[END_REF]Theorem 5.3.6] again, we get that lim

α sup x∈K g α (x) f (x) = sup x∈K g(x) f (x) .
Since f was chosen arbitrarily, we see that g α converges in the horofunction boundary to a point of the required form. , for all f ∈ C + (K), [START_REF] Roger | Hilbert's projective metric and iterated nonlinear maps[END_REF] where g : K → [1, ∞] is a lower-semicontinuous function with infimum 1. All these horofunctions are Busemann points.

Proof. Use Proposition 9.1 and the fact that the pointwise reciprocal map is an isometry taking the reverse-Funk metric to the Funk metric. Proof. By Theorem 8.2, the Busemann points of the Hilbert geometry are exactly the functions of the form ξ R + ξ F , with ξ R ∼ ξ F , where ξ R and ξ F are Busemann points of, respectively, the reverse-Funk and Funk geometries. The Busemann points of these geometries were described in Propositions 9.1 and 9.2. Let g and f be as in those propositions. Proposition 8.6 states that ξ R ∼ ξ F if and only if ḡ and f are not both positive and finite at any point of ca + r (K)\{0}. It is not to hard to show that this condition is equivalent to g and f not being both positive and finite at any point of K.

We have seen that all reverse-Funk horofunctions and all Funk horofunctions on the cone C + (K) are Busemann points. However, it is not necessarily true that all Hilbert horofunctions on this cone are Busemann. Indeed, consider the case where K := [0, 1] and take for example g := I [0,1/2) /2 + I [1/2,1] and f := I [0,1/2] + 2I (1/2,1] . By the propositions above, ξ R is a Busemann point of the reverse-Funk geometry and ξ F is a Busemann point of the Funk geometry, where ξ R and ξ F are defined as in [START_REF] Lemmens | Isometries of polyhedral Hilbert geometries[END_REF] and [START_REF] Roger | Hilbert's projective metric and iterated nonlinear maps[END_REF], respectively. Observe that if h n is a non-increasing sequence of continuous functions on C + (K) that converges pointwise to g, then h n is an almost-geodesic and converges to g in the reverse-Funk geometry; see Figure 2. Moreover, it converges to ξ F in the Funk geometry, although it is not an almost geodesic in this geometry. This shows that ξ R + ξ F is a horofunction of the Hilbert geometry. However, ξ R + ξ F is not a Busemann of this geometry, according to Proposition 9.3, since f and g do not satisfy the compatibility condition.

Thus, the situation differs from the finite-dimensional case. There, the reverse-Funk horofunctions are all automatically Busemann, and every Hilbert horofunction is Busemann if and only if every Funk horofunctions is.

The horofunction boundary of the Thompson geometry

Here we study the boundary of the Thompson geometry. Recall that reverse-Funk Busemann points are of the form (10 and Funk Busemann points are of the form (11. It was shown in Proposition 8.6, that if ξ R and ξ F are Busemann points of their respective geometries, then ξ R ∼ ξ F if and only if, for each y ∈ C * \{0}, either g(y) = 0 or f (y) = ∞. Here, g and f are the functionals appearing in [START_REF] Gromov | Hyperbolic manifolds, groups and actions[END_REF] and [START_REF] Lemmens | Isometries of infinite dimensional Hilbert geometries[END_REF].

r x (•) := log M C (x, •) M C (x, b) and f x (•) := log M C (•, x) M C (b, x) .
Let ∨ and ∧ denote, respectively, maximum and minimum. We use the convention that addition and subtraction take precedence over these operators. We write x + := x∨0 and x -:= x ∧ 0. Let R := R ∪ {-∞, +∞}. Given two real-valued functions f 1 and f 2 , and c ∈ R, define

[f 1 , f 2 , c] := f 1 + c -∨ f 2 -c + . Observe that if c = ∞, then [f 1 , f 2 , c] = f 1 , whereas if c = -∞, then [f 1 , f 2 , c] = f 2 .
Let B R and B F be the set of Busemann points of the reverse-Funk and Funk geometries, respectively. Proposition 10.1. Let C be a cone giving rise to a complete Thompson geometry. The set of Busemann points of this geometry is

{r z | z ∈ C} ∪ {f z | z ∈ C} ∪ B R ∪ B F ∪ {[r, f, c] | r ∈ B R , f ∈ B F , r ∼ f, c ∈ R}.
Proof. Let ξ T be a Busemann point of the Thompson geometry, and let z α be an almostgeodesic net in C converging to it.

By taking subnets if necessary, we may assume that z α converges in both the Funk and reverse-Funk horofunction compactifications, to the limits ξ R and ξ F , respectively, and furthermore that d R (b, z α ) -d F (b, z α ) converges to a limit c in R. So, as α tends to infinity,

d T (b, z α ) -d R (b, z α ) → -c -, and d T (b, z α ) -d F (b, z α ) → c + . Therefore, d T (y, z α ) -d T (b, z α ) = d R (y, z α ) ∨ d F (y, z α ) -d T (b, z α ) = d R (y, z α ) -d R (b, z α ) + c -∨ d F (y, z α ) -d F (b, z α ) -c + → [ξ R , ξ F , c](y).
Consider the case when c < ∞. Let λ > 0 be such that c < 2 log λ. For α large enough,

d R (λb, z α ) -d F (λb, z α ) = d R (b, z α ) -d F (b, z α ) -2 log λ < 0,
and hence d T (λb, z α ) = d F (λb, z α ). Note also that d T (z α , z α ) ≥ d F (z α , z α ), for all α and α .

Let f 1 and f 2 be distinct elements of K. So, there is some y ∈ C such that f 1 (y) = f 2 (y). Setting x := Φ -1 (y), we get ϕ x (f 1 ) = ϕ x (f 2 ). This shows that ϕ(X) separates the points of K.

Recall that we have chosen the basepoint b of the cone so that b = Φ(0). Write x := Φ -1 (eb), where e is Euler's number. Let f ∈ K. Since f is the logarithm of a linear functional, f (αz) = log α + f (z) for all points z in the cone, and all α > 0. So, since f (b) = 0, we get f (eb) = 1. Hence, ϕ x (f ) = f (eb) = 1. We have shown that the constant function 1 is in the image of ϕ.

Applying the Stone-Weierstrass theorem to cl Im ϕ, we get that cl Im ϕ is dense in C(K), and so Im ϕ is dense in C(K).

Theorem 11.3. If a Thompson geometry is isometric to a Banach space, then the cone is linearly isomorphic to C + (K), for some compact Hausdorff space K.

Proof. Assume that Φ : X → C is an isometry from a Banach space (X, || • ||) to a Thompson geometry on a cone C. We choose the basepoint b of the cone so that b = Φ(0).

Let K be the pointwise closure of the set of Funk singletons of C. Each element f of K can be written f = log y| C , where y is in the weak* closure of the set of extreme points of the cross-section {y ∈ C * | y, b = 1} of the dual cone C * . Since it is a closed subset of a compact set, K is compact. By Corollary 10.3, the pull back f •Φ of each Funk singleton f is a singleton Busemann point of the Banach space X, and is therefore linear, by Corollary 3.6. It follows that f • Φ is linear for all f ∈ K.

Let ϕ be defined as in Lemma 11.2. By Lemma 11.2, the image of the Banach space X under ϕ is a uniformly dense subspace of C(K).

Let F denote the set of Funk singletons. Recall that the following formula for the Funk metric holds for an arbitrary cone (see [START_REF] Walsh | Gauge-reversing maps on cones, and Hilbert and Thompson isometries[END_REF]Proposition 4.4]):

d F (w, z) = sup f ∈F f (w) -f (z) , for w, z ∈ C.
So, the Thompson metric is given by

d T (w, z) = d F (w, z) ∨ d F (z, w) = sup f ∈F f (w) -f (z) , for w, z ∈ C.
The same formula holds with F replaced by K since the former is dense in the latter. We conclude that, for all x, y ∈ X,

||y -x|| = d T (Φ(x), Φ(y)) = sup f ∈K |f (Φ(x)) -f (Φ(y))| = sup f ∈K |ϕ x (f ) -ϕ y (f )|.
Therefore ϕ is an isometry from (X, || • ||) to C(K) with the supremum norm. But we have assumed that X is complete, and so its image under ϕ is complete. We conclude that this image is the whole of C(K).

We have shown that ϕ is an isometric linear-isomorphism from X to C(K). p) , for all f ∈ K.

Define the map Θ

:= exp •ϕ • Φ -1 from C to C + (K). For each p ∈ C, we have (Θp)(f ) = exp •ϕ Φ -1 p (f ) = e f (
Since f is the logarithm of a linear functional on C, it follows that Θ is linear. So, Θ is a linear isomorphism between C and C + (K).

It is not hard to verify that this order makes D into a directed set. Define the net y β := x, for β := (W, x, n, ε, A). It is clear that y β converges to s. Suppose some λ > 0 is given. If β := (W, x, n, ε, A) is large enough that ε < λ/2, and β := (W , x , n , ε , A ) is such that β ≤ β , then combining (iii) and ( 14 This proves that the net y β is an almost geodesic.

Since s is linear, s (y β ) = s (z α ) + ns (v), for all β, where α is as in (i). Both α and n can be made as large as we wish by taking β large enough, and so s (y β ) converges to -∞.

Observe that if r and f are reverse-Funk and Funk horofunctions, respectively, then r + f is constant on each projective class of the cone, so we may consider it to be defined on P (C).

Lemma 12.4. Let Φ : X → P (C) be an isometry from a Banach space to the Hilbert geometry on a cone. Let s be a singleton of the Banach space. Write s = (r + f ) • Φ and -s = (r + f ) • Φ, where r and r are singletons of the reverse-Funk geometry, and f and f are singletons of the Funk geometry. Then, r = -f and r = -f . Proof. Let p and q be points in the projective space P (C) of the cone.

Let z β be an almost geodesic in the Banach space converging to s. Taking the reflection z β := 2Φ -1 (p) -z β in the point Φ -1 (p), we get an almost geodesic converging to -s.

Let y β := Φ(z β ), for all β. This is an almost geodesic in the Hilbert geometry, and so it is also an almost geodesic in both the reverse-Funk and Funk geometries. Hence, it converges to a Busemann point in both of these geometries. Moreover, the sum of the two Busemann points is equal to the Hilbert geometry Busemann point r + f . Since, by Theorem 8.2, a Hilbert geometry Busemann point can be written in a unique way as the sum of a reverse-Funk Busemann point and a Funk Busemann point, we see that the limit of y β in the reverse-Funk geometry is r and that the limit of y β in the Funk geometry is f . Likewise, let y β := Φ(z β ), for all β. Again, this is an almost geodesic in the Hilbert geometry, converging this time to r in the reverse-Funk geometry and to f in the Funk geometry.

For all β, since Φ -1 (p) is the midpoint of z β and z β , we have Combining these two equations with [START_REF] Walsh | Gauge-reversing maps on cones, and Hilbert and Thompson isometries[END_REF], and using the Funk metric triangle inequality applied to points y β , p, and y β , we get f (q) -f (p) + r (q) -r (p) ≥ 0. But this holds for arbitrary p and q in P (C), and so r + f must be constant on P (C). Since this function takes the value zero at b, we see that it is zero everywhere.

The proof that r + f = 0 is similar.

Given a cone C, we use F to denote the set of singletons f of the Funk geometry on C such that there exists a reverse-Funk singleton r satisfying r ∼ f . Lemma 12.5. Let C be a cone whose Hilbert geometry is isometric to a Banach space. Let Φ : X → C be a bijection from a linear space to C, such that the pullback h • Φ of every Hilbert geometry singleton h is a linear functional on X . If there exists f 0 ∈ F whose pullback is linear, then the pullback of every element of F is linear.

Proof. Since f 0 is in F , there is some reverse-Funk singleton r 0 satisfying r 0 ∼ f 0 , and so h 0 := r 0 + f 0 is a Hilbert singleton, by Corollary 8.9. By assumption, s := h 0 • Φ is linear. We have also assumed that f 0 • Φ is linear, so we deduce that r 0 • Φ is linear.

Let f ∈ F . So, there is some singleton r of the reverse-Funk geometry such that h := r + f is a singleton of the Hilbert geometry. From the isometry between the Banach space and the Hilbert geometry, we get that there is another singleton h of the Hilbert geometry satisfying h = -h. We may write h = r + f , where r and f are singletons of the reverse-Funk and Funk geometries, respectively.

Using Lemma 12.3 and the isometry between the Banach space and the Hilbert geometry, we get that there exists an almost geodesic net [y α ] in P (C) converging in the Hilbert geometry to h 0 such that either h(y α ) or h (y α ) converges to -∞.

Consider the former case. For all α, take a representative y α of the projective class [y α ] so that d F (b, y α ) = 0. This implies, for each α, that •, b ≤ •, y α on C, from which it follows that f (y α ) ≥ 0. Since we are supposing that h(y α ) = r(y α ) + f (y α ) converges to -∞, we must have that r(y α ) converges to -∞. From Proposition 8.1, we get that y α is an almost geodesic in the Funk geometry. In this geometry, it converges to f 0 . Applying Lemma 12.1, we get that r ∼ f 0 . This means that r + f 0 is a singleton of the Hilbert geometry on C, and so, by assumption, its pullback (r + f 0 ) • Φ is linear. We deduce that r • Φ is linear, and hence that f • Φ = (h -r) • Φ is linear. Now consider the second case, that is, where h (y α ) = r (y α ) + f (y α ) converges to -∞. Using reasoning similar to that in the previous paragraph, we get that r • Φ is linear. But, according to Lemma 12.4, r = -f , and so f • Φ is linear.

Let K be a compact space. Define the following seminorm on C(K).

||x|| H := sup f ∈K x(f ) -inf f ∈K x(f ).
Denote by ≡ the equivalence relation on C(K) where two functions are equivalent if they differ by a constant, that it, x ≡ y if x = y + c for some constant c. The seminorm ||x|| H is a norm on the quotient C(K)/≡. This space is a Banach space, and we denote it by H(K).

Theorem 12.6. If a Hilbert geometry on cone C is isometric to a Banach space, then C is linearly isomorphic to C + (K) for some compact Hausdorff space K.

Proof. Let Φ : X → P (C) be an isometry from a Banach space (X, || • ||) to the Hilbert geometry on a cone C.

Each singleton of the Hilbert geometry may be written r + f , where r and f are singletons of the reverse-Funk and Funk geometries, respectively. Let F denote the set of Funk singletons that appear in this way, and denote by K the pointwise closure of this set of functions.

Recall that each element f of K can be written f = log y| C , where y is in the weak* closure of the set of extreme points of the cross-section {y ∈ C * | y, b = 1} of the dual cone C * . Since K is a closed subset of a compact set, it is compact.

Consider the linear space X := X × R, and define a map Φ : X → C in the following way. Fix a choice of a particular f 0 ∈ F . For each x ∈ X, the projective class Φ(x) is a ray in C, and along this ray the function f 0 is monotonically increasing, taking values from -∞ to ∞. So, for each x ∈ X and α ∈ R, we may define Φ (x, α) to be the representative of Φ(x) where f 0 takes the value α.

Observe that, since by definition (f 0 • Φ )(x, α) = α, for all x ∈ X and α ∈ R, [START_REF] Walsh | The horofunction boundary of finite-dimensional normed spaces[END_REF] we have that f 0 • Φ is linear.

Consider the pull back h • Φ of a Hilbert singleton h. This is a function on X that is independent of the second coordinate. Moreover, the dependence on the first coordinate is linear since each singleton of the Banach space X is linear. So, h • Φ is linear on X .

Applying Lemma 12.5, we get that f • Φ is linear for all f ∈ F , and it follows that the same is true for all f ∈ K.

Define the map ϕ : X → C(K), x → ϕ x as in Lemma 11.2. That is, ϕ x (f ) := f (Φ (x)), for all f ∈ K.

According to Lemma 11.2, ϕ is linear and its image is a uniformly dense subspace of C(K).

We make the following claim, the proof of which we postpone.

Claim 1. The image of X under the map ϕ is a complete subset of (C(K), || • || ∞ ).

Using this claim, we conclude that the image of ϕ is the whole of C(K).

Extend the norm on X to a seminorm on X by ignoring the second coordinate. We again use || • || to denote this seminorm. We make a second claim. It follows from this, upon quotienting on each side by the respective subspace where the seminorm is zero, that X is isometric to H(K).

Define the map Θ := exp •ϕ • Φ -1 from C to C + (K). For each p ∈ C, we have (Θp)(f ) = exp •ϕ Φ -1 p (f ) = e f (p) , for all f ∈ K.

Since f is the logarithm of a linear functional on C, it follows that Θ is linear. So, Θ is a linear isomorphism between C and C + (K).

Proof of Claim 2. Let S be the set of singletons of the Banach space (X, || • ||). Extend each s ∈ S to X by s(x, α) := s(x), for all x ∈ X and α ∈ R.

We have that ||p|| = sup s∈S s(p), for all p in X. We have seen that a function s is in S if and only if it is of the form s = f • Φ -f • Φ , with f and f distinct elements of F . Note that if f are f were identical, then f • Φ -f • Φ would be zero. So, for any p ∈ X ,

||p|| = sup f,f ∈F f • Φ (p) -f • Φ (p) = sup f ∈F ϕ p (f ) -inf f ∈F ϕ p (f ) = ||ϕ p || H , since F is dense in K.

Figure 1 .

 1 Figure 1. Hilbert's definition of a distance.

  d H (x, y) := log |zx| |wy| |zy| |wx| .

  This order makes D into a directed set. For each α := (W, x) ∈ D, let z α := x. Clearly, the net z α converges to ξ. Let E be an open neighbourhood of H(ξ, η) in [0, ∞], and let (W , x ) ∈ D. Take W ∈ N such that W ⊂ W and inf y∈W ∩X f (y) ∈ E.

Lemma 5 . 3 .

 53 The function Ξ in (6) is lower semicontinuous.

Lemma 6 . 2 .

 62 Let D be a compact convex subset of a locally-convex Hausdorff space E, and let f 1 and f 2 be lower-semicontinuous positive affine functions on D. If sup D g/f 1 ≤ sup D g/f 2 for each continuous real-valued positive affine function g on D, then f 1 ≥ f 2 on D.

Corollary 7 . 3 .

 73 The two Funk Busemann points ξ 1 and ξ 2 are in the same part if and only iff 2 /λ ≤ f 1 ≤ λf 2 , for some λ > 0.Unlike in the case of the reverse-Funk geometry, we can determine explicitly the singleton Busemann points of the Funk geometry. Recall that we have defined the cross section D := {y ∈ C * | y, b = 1}. Corollary 7.4. A function ξ is a singleton Busemann point of the Funk geometry if and only if it is an extreme point of D.

Lemma 8 . 3 .

 83 We have f = sup{h | (h, h ) ∈ C} and g = inf{h | (h, h ) ∈ C}.Proof. Denote by D the cross-section of the dual cone C * , and by E the affine hull of D.Define the epigraph of f and the (truncated) hypograph of g:

9. 1 .

 1 The boundary of the reverse-Funk geometry on C + (K). The reverse-Funk metric on C + (K) is given byd R (f, g) = log sup x∈K g(x) f (x), for all g, f in C + (K).

Proposition 9 . 1 .

 91 of functions is said to converge in the hypograph topology if their hypographs converge in the Kuratowski-Painlevé topology. The horofunctions of the reverse-Funk geometry on the positive cone C + (K) are the functions of the form ξ R (f ) := log sup x∈K g(x) f (x)

9. 2 .Proposition 9 . 2 .

 292 The boundary of the Funk geometry on C + (K). The horofunctions of the Funk geometry on the positive cone C + (K) are the functions of the form ξ F (f ) := log sup x∈K f (x) g(x)

9. 3 .

 3 The boundary of the Hilbert geometry on C + (K).Proposition 9.3. The Busemann points of the Hilbert geometry on the positive cone C + (K) are the functions of the formξ H (h) := log sup x∈K h ∈ C + (K),where g : K → [0, 1] is an upper-semicontinuous function with supremum 1, and f : K → [1, ∞] is a lower-semicontinuous function with infimum 1, and, for each x ∈ K, either g(x) = 0 or f (x) = ∞.

Figure 2 .

 2 Figure 2. A sequence converging to a horofunction

  ) we get d(b, y β ) + d(y β , y β ) -d(b, y β ) < λ.

d

  H (y β , p) + d H (p, y β ) = d H (y β , y β ). This implies that d F (y β , p) + d F (p, y β ) = d F (y β , y β ), for all β. (15)Recall thatf (q) -f (p) = lim β d F (q, y β ) -d F (p, y β ) and r (q) -r (p) = lim β d F (y β , q) -d F (y β , p) .

Claim 2 .

 2 The map ϕ is an isometry between (X , || • ||) and (C(K), || • || H ).

Recall that an almost-geodesic remains an almost-geodesic when the basepoint is changed. It will be convenient to consider almost-geodesics with respect to the basepoint λb.

Let ε > 0 be given. Since z α is an almost geodesic in the Thompson geometry, we have, for α and α large enough, with α < α , d T (λb, z α ) ≥ d T (λb, z α ) + d T (z α , z α ) -ε, and hence

We deduce that z α is an almost-geodesic in the Funk geometry, and so ξ F is either of the form f z , with z ∈ C, or a Funk Busemann point.

Similarly, when c > -∞, z α is an almost-geodesic in the reverse-Funk geometry and ξ R is either of the form r z , with z ∈ C, or a reverse-Funk Busemann point.

There remains the case where c is finite. Since ξ T was assumed to be in the horofunction boundary, we have, by Proposition 2.6, that d T (b, z α ) converges to infinity, and so both d R (b, z α ) and d F (b, z α ) do too, since their difference remains bounded. Therefore both ξ R and ξ F are Busemann points. We have shown that

We extended the definition of H by setting H(ξ + u, η + v) := H(ξ, η) + v -u for all Busemann points ξ and η, and u, v ∈ [-∞, 0]. Here, we use the convention that -∞ is absorbing for addition. The following proposition was proved in [START_REF] Walsh | Gauge-reversing maps on cones, and Hilbert and Thompson isometries[END_REF] in the finitedimensional case, but the proof carries over to infinite dimension.

Proposition 10.2. The distance in the detour metric between two Busemann points ξ T and ξ T in a complete Thompson geometry is δ(ξ T , ξ T ) = d H (x, x ) if ξ T = r x and ξ T = r x , with x, x ∈ D. The same formula holds when ξ T = f x and ξ

Corollary 10.3. The set of singletons of a complete Thompson geometry is exactly the union of the Funk singletons and the reverse-Funk singletons.

Thompson geometries isometric to Banach spaces

In this section we determine which Thompson geometries are isometric to Banach spaces.

We start with a technical lemma.

Lemma 11.1. For all α, β ∈ R, the sequence p n := n -1 log(exp(nα) + exp(nβ)) is non-increasing.

Proof. Fix n ∈ N. Observe that, for any r ∈ {0, . . . , n}, e (α-β)(n-r) + e (β-α)r ≥ 1, since one or other of the terms is greater than or equal to 1. Equivalently, e αn + e βn ≥ e αr e β(n-r) .

By considering binomial coefficients and using the previous inequality, we get that (e αn + e βn )(e αn + e βn ) n ≥ e α(n+1) + e β(n+1) n

Taking logarithms and rearranging, we get p n ≥ p n+1 .

Recall that a linear subspace of a Riesz space (vector lattice) E is a Riesz subspace if it is closed under the lattice operations on E. We will need the lattice version of the Stone-Weierstrass theorem. This theorem states [START_REF] Aliprantis | Principles of real analysis[END_REF] that if K is a compact space, then any Reisz subspace of C(K) that separates the points of K and contains the constant function 1 is uniformly dense in C(K).

The setting for the next lemma is an order unit space (V, C, b).

Lemma 11.2. Let K be the pointwise closure of a set of Funk singletons of the cone, and let Φ be a bijection from a linear space X to C, such that the pullback f • Φ of each element f of K is linear. Then, the map ϕ : X → C(K), x → ϕ x , where

is linear and its image is dense in C(K).

Proof. Observe that, as a closed subset of a compact set, K is compact.

We have, for all α, β ∈ R, and x, y ∈ X, and f ∈ K,

Therefore, ϕ is linear. Furthermore, the image of ϕ is a linear subspace of C(K).

Define the following operation on X:

x ⊕ n y := 1 n Φ -1 (Φ(nx) + Φ(ny)), for x, y ∈ X.

Fix x and y in X. Let f ∈ K, and write g := f • Φ. So, g is a linear functional on X.

Recall that f is the logarithm of a linear functional l on the cone, that is, f = log • l for some linear functional l on C. Note that l = exp • f . We have

Using Lemma 11.1, we get that the sequence g(x ⊕ n y) = ϕ x⊕ny (f ) converges monotonically to its limit, max{g(x), g(y)}, as n tends to infinity. Since this is true for every f ∈ K, we have that ϕ x⊕ny converges monotonically and pointwise to ϕ x ∨ ϕ y , and hence converges to this limit uniformly, by Dini's theorem. Therefore, ϕ x ∨ ϕ y is in cl Im ϕ, the closure of the image of ϕ. It follows that cl Im ϕ is a Reisz subspace of C(K).

Hilbert geometries isometric to Banach spaces

Here we prove that the only Hilbert geometries isometric to Banach spaces are the ones on the cones C + (K), for some compact space K.

We first require some lemmas concerning singleton Busemann points in cones geometries and in Banach spaces.

Lemma 12.1. Let z α be an almost-geodesic net in the Funk geometry on a cone C, converging to a Funk Busemann point ξ F and normalised so that

Proof. Write ξ R and ξ F in the form [START_REF] Gromov | Hyperbolic manifolds, groups and actions[END_REF] and [START_REF] Lemmens | Isometries of infinite dimensional Hilbert geometries[END_REF], respectively, for some functionals g and f .

Choose ε > 0. Since z α is a Funk almost-geodesic, normalised in the way we have assumed, we have, for α large enough, that

Applying Lemma 6.2, we get that •, z α < exp(ε)f , for α large enough. Since ξ R (z α ) converges to -∞, we have that sup y∈C * \{0} g(y)/ y, z α converges to zero, which implies that g(y)/ y, z α converges to zero for all y ∈ C * \{0}. We deduce that, if g(y) is non-zero for some y ∈ C * \{0}, then y, z α converges to infinity, and so f (y) is infinite. The conclusion follows on applying Proposition 8.6.

Lemma 12.2. Let s be a singleton of a Banach space, and let ϕ : x → x + v be the translation by some vector v. Then, ϕ leaves s invariant.

Proof. By Corollary 3.6, s is linear, and the conclusion follows easily.

Lemma 12.3. Let s and s be singletons of a Banach space X. Then, there exists an almost-geodesic net z α converging to s such that s (z α ) converges to either ∞ or -∞.

Proof. Let z α be an almost geodesic converging to s, and denote by D be the directed set on which the net z α is based. By taking a subnet if necessary, we may assume that s (z α ) converges to a limit in [-∞, ∞]. If this limit is infinite, then the conclusion of the lemma holds, so assume the contrary.

Take a point v in the Banach space such that s (v) < 0. Observe that for each n ∈ N, the net z α + nv is an almost geodesic, and by Lemma 12.2 it converges to s.

We denote by d(•, •) the metric coming from the norm || • ||. Denote by N the set of neighbourhoods of s in the horofunction compactification of X. Let D be the set of elements