ISSN 0001-4346, Mathematical Notes, 2017, Vol. 101, No. 2, pp. 193–202. © Pleiades Publishing, Ltd., 2017. Original Russian Text © P. D. Andreev, 2017, published in Matematicheskie Zametki, 2017, Vol. 101, No. 2, pp. 169–180.
Normed Space Structure on a Busemann G-Space of Cone Type P. D. Andreev* Lomonosov Northern (Arctic) Federal University, Arkhangelsk, Russia Received June 3, 2016
Abstract—It is proved that any Busemann nonpositively curved G-space of cone type is isometric to a finite-dimensional normed space with strictly convex norm. DOI: 10.1134/S0001434617010230 Keywords: Busemann G-space, tangent cone.
1. INTRODUCTION The notion of a G-space was introduced by Busemann in a 1940s series of papers (including [1], [2], and other papers). A detailed review of the theory of G-spaces is contained in the book [3], which considers, in particular, nonpositively curved G-spaces. In [4], it was shown that any Busemann nonpositively curved G-space is a topological manifold. This proves Busemann’s conjecture that any G-space is a topological manifold for the class of nonpositively curved G-spaces. The main tool in the proof of the theorem cited above is the tangent cone. In [4], it was shown that, for any G-space X with globally nonpositive curvature, the tangent cone Kp X to X with vertex at any point p ∈ X is homeomorphic to X and has the same properties, i.e., is a nonpositively curved G-space. Moreover, the group of positive homotheties with center p acts on Kp X. The rest of [4] is devoted to the study of properties of a cone-type G-space with vertex p, i.e., a nonpositively curved G-space X for which the cone Kp X is isometric to X. A G-space is of cone type with vertex p if and only if the group H of positive homotheties with center p acts on X. In this paper, we significantly strengthen results on the metric structure of cone-type G-spaces. Our main result is the following theorem. Theorem 1. Let (X, d) be a Busemann nonpositively curved G-space of cone type with vertex p ∈ X. Then X is isometric to a finite-dimensional vector space V endowed with a strictly convex norm. Theorem 1 makes it possible to substantially shorten the proof of the main theorem in [4]: after the consideration of properties of the tangent cone to X, the necessity for further investigation is virtually eliminated. This paper is organized as follows. In the next section, we give the necessary definitions and facts from the theory of Busemann nonpositively curved G-spaces. Section 3 contains the proof of Theorem 1, which consists in proving a series of auxiliary assertions and is completed by inductively proving a general fact which implies that the convex hull of a k-dimensional normed subspace of X and a straight line not contained in this subspace is a (k + 1)-dimensional normed subspace. This fact and the finite compactness of G-spaces automatically imply the required result. *
E-mail:
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2. PRELIMINARIES Let (X, d) be a metric space. By the segment in X joining points x and y, which are the endpoints of the segment, we mean the image in X of a rectifiable path of length d(x, y). In other words, a segment in X in an isometric image of a numerical interval [a, b] ⊂ R. A segment with endpoints x, y ∈ X does not generally exist, and if such a segment exists, then it may be determined not uniquely. Similarly, by a line in X we mean the isometric image of the real number line R, and by a ray we mean the isometric image of the number half-line R+ = [0, +∞). The points of a segment which are not its endpoints are said to be interior. Let x, y, z ∈ X be three different points. We say that a point z is between x and y if d(x, y) = d(x, z) + d(z, y). In this case, we write x − z − y. In particular, all interior points of any segment with endpoints x and y are between x and y. A space X is said to be geodesic if any two points can be joined by a segment in X. Lines a, b : R → X in a geodesic space X are said to be parallel if the Hausdorff distance between them is finite: Hd(a, b) < +∞. The Hausdorff distance Hd(A, B) between closed subsets A, B ⊂ X is defined by Hd(A, B) = inf ε (∀ x ∈ A ∃ y ∈ B : d(x, y) ≤ ε) ∧ (∀ y ∈ B ∃ x ∈ A : d(x, y) ≤ ε) . In particular, the parallelism relation between lines is reflexive: any line a is parallel to itself. A metric space X is called a Busemann G-space if the following axioms hold: (1) X is finitely compact, i.e., any bounded closed subset of X is compact; (2) X is Menger convex, i.e., given any two points x, y ∈ X, there exists a point z ∈ X between them; (3) the segments in X are locally extendable: given any point x ∈ X, there exists a number rx > 0 such that, whenever y, z ∈ U (x, rx ), there is a w ∈ X for which y − z − w; (4) the extension of each segment in X is unique: if x − y − u and x − y − v, then either y − u − v or y − v − u. The finite compactness and Menger convexity of X imply that X is geodesic, and axioms (3) and (4) imply that, for any point x ∈ X and any y, z ∈ U (x, rx ), the segment joining y and z is determined uniquely. In [5], Busemann defined the property of having nonpositive curvature for G-spaces, which he regarded as a key property in his subsequent work. Definition 1. A geodesic space (X, d) is said to be Busemann nonpositively curved if, for any three points x, y, z ∈ X, the midpoint m between x and y and the midpoint n between x and z satisfy the condition 1 d(m, n) ≤ d(y, z). 2 In other words, in a Busemann nonpositively curved space, the length of the medial line of any triangle is at most half the length of the corresponding base. The simplest consequences of the nonpositivity of curvature are the following properties. Any nonpositively curved space X is contractible, and any two points y, z ∈ X are joined by a unique segment, which we denote by [yz]. Also, importantly, the metric d is convex in this case: for any two segments [uv] and [yz] with respective parameterizations γ, δ : [0, 1] → X proportional to natural parameterizations, the function d(γ(s), δ(t)) is a convex function of two variables (s, t) ∈ [0, 1] × [0, 1]. For this reason, Busemann nonpositively curved spaces are sometimes referred to as convex spaces. The notion of a Busemann nonpositively curved space also has a local version, in which curvature is nonpositive only locally rather than in the entire space. Each point of a locally Busemann nonpositively curved space has a neighborhood in which any three points x, y, and z satisfy the condition in MATHEMATICAL NOTES
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Definition 1. In what follows, we assume that X is a space with globally nonpositive curvature in the sense of Definition 1. An important tool in the theory of nonpositively curved spaces is the following lemma of Rinow (see [6], [7]). Lemma 1. Any two parallel lines a and b in a Busemann nonpositively curved space X bound a normed strip, i.e., a convex subset of X isometric to a strip between two parallel straight lines in an affine plane endowed with a strictly convex norm. An important property of the geometry of parallel lines in a nonpositively curved space was proved in [8, Lemma 2.7]. Let a ⊂ X be a line. We use (Ya , Hd) to denote the metric space formed by the set Ya of lines parallel to a in X with Hausdorff metric Hd. Lemma 2. Let (X, d) be a Busemann nonpositively curved space. Then the space (Ya , Hd) is Busemann nonpositively curved as well. Rays a, b : R+ → X in a geodesic space X are said to be asymptotic if the Hausdorff distance between them is finite: Hd(a, b) < +∞. Asymptoticity is an equivalence relation on the set of rays in X. The equivalence classes form the so-called geodesic boundary ∂g X of the space X. If X is a Busemann nonpositively curved finitely compact space and q ∈ X, then, for any ray c : R+ → X, an asymptotic ray with origin q is uniquely determined. The geodesic boundary of a Busemann nonpositively curved finitely compact space X and its geodesic compactification X g = X ∪ ∂g X was studied in [9]. The points of the geodesic boundary are regarded as infinite. The infinite point determined by a ray a is denoted by a(+∞). Any line c : R → X determines two infinite points, c(−∞) and c(+∞), which correspond to two opposite rays of this line. In [4], a tangent cone to a nonpositively curved G-space was defined. Below we give the necessary facts related to this notion. Definition 2. Let (X, d) be a Busemann nonpositively curved G-space with a marked point p ∈ X. For each t ≥ 1, we define a metric dt on X by dt (x, y) = t · d(xt , yt ), where xt ∈ [px] and yt ∈ [py] are points for which d(p, x) = t · d(p, xt ) and d(p, y) = t · d(p, yt ). Then the limit d∗ (x, y) = lim dt (x, y) t→+∞
exists, and the function d∗ is a metric on X. The metric space (X, d∗ ) is called the tangent cone to X with vertex p and denoted by Kp X. Properties of the tangent cone are listed in the following theorem [4, Theorem 4]. Theorem 2. Let (X, d) be a Busemann nonpositively curved G-space. Then the tangent cone Kp X = (X, d∗ ) is a Busemann nonpositively curved G-space as well, and the following assertions hold: • d∗ (y, z) ≤ d(y, z) for any y, z ∈ X; • the map Id : X → X is a homeomorphism between (X, d) and Kp X; • the lines through p in the sense of the metric d∗ coincide with similar lines in the sense of d, and d = d∗ on each of these lines; • the group H of positive homotheties with center p acts on the space Kp X. MATHEMATICAL NOTES
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The objects of study in this paper are Busemann nonpositively curved G-spaces of cone type. Such spaces include, in particular, the tangent cone Kp X = (X, d∗ ) of X. Definition 3. A Busemann nonpositively curved G-space (X, d) is called a space of cone type with vertex p ∈ X if the identity map Id : X → Kp X is an isometry. A criterion for a space X to be of cone type with vertex p is the action on X of the group of positive homotheties with center p [4, Lemma 9]. Lemma 3. A Busemann nonpositively curved G-space (X, d) is of cone type with vertex p ∈ X if and only if the group H of positive homotheties with center p acts on X. 3. PROOF OF THE MAIN RESULT Throughout the rest of this paper, X is a Busemann nonpositively curved G-space of cone type with vertex p. The following technical lemmas and their corollaries lead step by step to the main result. Lemma 4. Let a, b : R+ → X be natural parameterizations of rays a, b ⊂ X with common origin a(0) = b(0) = p which are not opposite to each other. Suppose that points x, u ∈ a and y, v ∈ b are such that p − x − u and p − v − y. Then the segments [xy] and [uv] have a common point. Proof. We set α=
d(p, v) < 1, d(p, y)
β=
d(p, u) > 1, d(p, x)
λ=
α(β − 1) < 1. β−α
(3.1)
Let m ∈ a and n ∈ b be points for which d(p, u) = α, d(p, m)
d(p, v) = β. d(p, n)
The segment [uv] is the image of [my] under the homothety hα and of [xn] under the homothety hβ . Therefore, d(u, v) = α · d(m, y) = β · d(x, n). Choose a point w on [xy] so that d(x, w) = λ. d(x, y)
(3.2)
It is easy to show that d(x, u) = λ. d(x, m) Since the metric is convex, it follows that d(u, w) ≤ λ · d(m, y), and since β(1 − α) d(y, v)) = = 1 − λ, d(y, n) β−α it follows that d(w, v) ≤ (1 − λ) · d(x, n). We obtain d(u, v) ≤ d(u, w) + d(w, v) ≤ λ · d(m, y) + (1 − λ) · d(x, n) λ 1−λ + d(u, v) = d(u, v). = α β The coincidence of the left- and right-hand sides implies that each of the nonstrict inequalities in this chain is an equality. In particular, d(u, v) = d(u, w) + d(w, v). This and the uniqueness of the segment [uv] ⊂ X imply w ∈ [uv]. MATHEMATICAL NOTES
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The following assertion is easy to prove on the basis of the proof of Lemma 4. Lemma 5. Suppose that rays a, b : R+ → X and points x ∈ a and y ∈ b satisfy the same assumptions as in Lemma 4; suppose also that w ∈ [xy] and d(p, x) < d(p, y). Then there exist points x ∈ a and y ∈ b for which d(p, x) ≤ d(p, x ) = d(p, y ) ≤ d(p, y)
and
w ∈ [x y ].
Proof. The quantity λ defined by (3.1) continuously depends on α and β, which are determined by the distance d = d(p, y ) = d(p, x ) (x and y are the required points) and continuously depend on d. As d varies from d(p, x) to d(p, y), the parameter λ varies from 0 to 1. Hence there exists a value of d such that the corresponding value of λ satisfies (3.2). It follows from the proof of Lemma 4 that w is a common point of the segments [xy] and [x y ]. Lemma 6. Suppose that rays a, b : R+ → X are the same as in Lemma 4, x ∈ a, y ∈ b, and w ∈ [xy] is any point. Then, for any u ∈ a and v ∈ b, the segment [uv] intersects the ray [pw). Proof. By virtue of Lemma 5 and the action of the group H on X, it suffices to consider fixed points x and y for which d(p, x) = d(p, y) = 1; moreover, we can assume that d(p, u) ≥ d(p, v). Let λ=
d(x, w) . d(x, y)
If d(p, u) = d(p, v), then where
[uv] = hα ([xy]),
α=
d(p, u) , d(p, x)
and the ray [xw) intersects [uv] in the point w = hα (w). Suppose that d(p, u) > d(p, v). In this case, we set x = hσ (x), y = hσ (y), and w = hσ (w) ∈ [x y ], where d(p, u)d(p, v) . σ= (1 − λ)d(p, v) + λd(p, u) For the numbers α < 1 and β > 1 defined by α=
d(p, v) d(p, v) = , (p, y ) σ
β=
d(p, u) d(p, u) = , d(p, x ) σ
we have d(p, v)(d(p, u) − σ) α(β − 1) = = λ. β−α σ(d(p, u) − d(p, v)) It follows from the proof of Lemma 4 that w is a common point of the segments [x y ] and [uv]. Therefore, w = [pw) ∩ [uv]. Corollary 1. Suppose that rays a, b : R+ → X are the same as in Lemma 4, x ∈ a and y ∈ b are any points, and ConvHull(a ∪ b) is the convex hull of the union of the rays a and b. Then hk ([xy]) = [pw). (3.3) ConvHull(a ∪ b) = {p} ∪ k>0
w∈[xy]
is convex, whence [pw). ConvHull(a ∪ b) ⊆
Proof. It follows from Lemma 6 that the set
w∈[xy] [pw)
w∈[xy]
We obtain {p} ∪
hk ([xy]) ⊆ ConvHull(a ∪ b) ⊆
k>0
[pw) ⊆ {p} ∪
w∈[xy]
The first inclusion is obvious, and the last follows from Lemma 3. MATHEMATICAL NOTES
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hk ([xy]).
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Corollary 2. Suppose that a, b : R+ → X are the same rays as in Lemma 4. Then the convex hull ConvHull(a ∪ b) is closed in X. Proof. The set determined by the first equality in (3.3) is closed. Indeed, let D = d(x, y), and let ϕ : R+ × [0, D] → X be the map such that ϕ(0, t) = p, ϕ1 (t) = ϕ(1, t) is a natural parameterization of the segment [xy], and ϕ(s, t) = hs (ϕ1 (t)). It is easy to see that ϕ is continuous. Therefore, the image ϕ([0, s] × [0, D]) is compact and, hence, closed for any s > 0. Let m = d(p, [xy]) = min d(p, ϕ1 (t)). t∈[0,D]
Then d(p, hs ([xy]) = sm. If a point z belongs to the closure {p} ∪ k>0 hk ([xy]), then, for any ε > 0, the ε-neighborhood of z does not intersect ϕ((S, +∞), [0, D]), where d(p, z) + ε . m Therefore, z belongs to the closure of the set ϕ([0, d(p, z)/m] × [0, D]) and, hence, to this set itself. S>
Corollary 3. Let a, b : R+ → X be the same rays as in Lemma 4, and let c : R+ → X be the ray in X with origin a(s), s > 0, asymptotic to b. Then the image of c is contained in ConvHull(a ∪ b). Proof. For any τ ≥ 0, the point c(τ ) is the limit as t → +∞ of the points dt (τ ), where dt : [0, d(a(s), b(t))] → X is a natural parametrization of the segment [a(s)b(t)]. We have dt (τ ) ∈ ConvHull(a, b), and we see that ConvHull(a, b) is closed in X; this implies the required assertion. The following assertion follows from the preceding ones. Corollary 4. The relation ConvHull(a ∪ b) =
cs
s≥0
holds, where c0 = b and cs : R+ → X is the ray with origin a(s), s > 0, asymptotic to b. For the convex hull ConvHull(a, b) of two rays a and b with origin p which are not opposite to each other, we use the notation (a, b) accentuating its geometry. Indeed, the properties of ConvHull(a, b) proved above show that, in fact, this is merely a sector with sides a and b. Lemma 7. Suppose that a line c : R → X does not pass through the vertex p of X. Then c is parallel to the line cλ = hλ (c) for any λ > 0. Proof. Consider the case 0 < λ < 1. Let c = c(t), where t ∈ R is any natural parameterization of the line c. Then cλ (t) := ht (c(t/λ)) is a natural parameterization of cλ . For any t, the point cλ (t) belongs to the segment [pc(t/λ)] and divides it in the ratio of λ/(1 − λ). The point c(t) divides the segment [c(0)c(t/λ)] in the same ratio. Applying the curvature nonpositivity property to these segments, we obtain the estimate d(c(t)cλ (t)) ≤ (1 − λ)d(p, c(0)) = d(cλ (0), c(0)). Therefore, the distances d(c(t)cλ (t)) are bounded above, which implies that the given lines are parallel. The case λ > 1 is reduced to the preceding one by setting d = cλ and dμ = c, where μ = 1/λ. MATHEMATICAL NOTES
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Lemma 8. Let a, b : R+ → X be the same rays as in Lemma 4, and let a− : R+ → X be the ray opposite to a. Then ConvHull(a ∪ a− ∪ b) = (a, b) ∪ (a− , b), and ConvHull(a ∪ a− ∪ b) is a strictly convex normed half-plane in X. Proof. Let c+ : R+ → X be the ray with origin b(1) asymptotic to a, and let c− be the opposite ray; we set c = c+ ∪ c− and cλ = hλ (c). Suppose that c− is not contained in (a− , b). Then the infinite points of c− (+∞) and a− (+∞) are different, while c+ (+∞) = a(+∞) and cλ (+∞) = c+ (+∞) = a(+∞),
cλ (−∞) = c− (+∞)
for all λ > 0. As λ → 0, the lines cλ converge to c0 . We have c0 (t) = a(t)
for t ≥ 0
and
c0 (−∞) = a− (∞).
This contradicts the uniqueness of the extension of segments beyond the point p. Thus, cλ ((−∞, 0]). (a− , b) = a− ∪ λ>0
Applying Lemma 1, we see that the union of the parallel lines cλ and the rays a and a− is a normed half-plane in X with strictly convex norm. Now, together with a pair of rays a+ , b+ : R+ → X with common origin p, consider the pair of rays a− and b− opposite to them. The union of a+ and a− is a line a : R → X. Similarly, b = b+ ∪ b− . The convex hull ConvHull(a ∪ b) can be represented in the form of the union of four sectors as ConvHull(a ∪ b) = (a+ , b+ ) ∪ (b+ , a− ) ∪ (a− , b− ) ∪ (b− , a+ ) and in the form of the union of four half-planes as ConvHull(a ∪ b) = ConvHull(a+ ∪ a− ∪ b+ ) ∪ ConvHull(b+ ∪ b− ∪ a− ) ∪ ConvHull(a− ∪ a+ ∪ b− ) ∪ ConvHull(b− ∪ b+ ∪ a+ ). We form a cycle from these four half-planes and note that every two consecutive half-planes in this cycle have a common sector. Taking each of the half-planes for a normed half-plane, we see that the norms of these half-planes coincide on their intersections. As a result, we obtain the following corollary of Lemma 8. Corollary 5. The convex hull of lines a, b ⊂ X intersecting in the vertex p is a normed plane. Now it is easy to prove the following strengthening of Lemma 2 in the case where X is a G-space of cone type. Lemma 9. Let (X, d) be a G-space nonpositively curved G-space of cone type with vertex p, and let a be a line passing through p. Then the space (Ya , Hd) is a nonpositively curved G-space of cone type as well. Finally, we proceed to the proof of the main theorem. First, we prove by induction that, for any positive integer n, the following assertion holds. Proposition 1. Let X be a nonpositively curved G-space X of cone type with vertex p, and let α : Rn → X be an n-dimensional normed plane in X passing through p. Then, for any point q ∈ X not contained in α, the convex hull of the plane α and the line pq is a normed (n + 1)-plane with strictly convex norm. MATHEMATICAL NOTES
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Proof. The base case n = 1 of the induction was proved in Corollary 5. Suppose that the required assertion holds for n = k − 1, i.e., is true for all nonpositively curved G-spaces of cone type and for any (k − 1)-dimensional normed plane is any such space X. Let us prove that this assertion is also true for n = k. Let α : Rk → X be any k-plane in X passing through the vertex p. Consider a line a ⊂ α in α containing p. For any point q ∈ X not belonging to a, the lines a and b = pq determine the normed 2-plane βq = ConvHull(a ∪ b). In the plane βq , we can draw a line cq parallel to a through the point q. We have cq . X =a∪ q ∈a /
Consider the space (Ya , Hd) of lines in X parallel to a (including a) with Hausdorff metric Hd. By Lemma 9, this is a nonpositively curved G-space of cone type whose vertex is the line a treated as an element of Ya . The subspace Ya,α ⊂ Ya consisting of all lines contained in α is isometric to the (k − 1)-dimensional quotient of Ya by a one-dimensional subspace collinear with a; we endow the quotient space with the corresponding quotient norm. Let q ∈ / α. Consider the line cq ∈ Ya containing q and the line Ya,βq in the space Ya formed by the elements of this space (that is, lines in X) contained in βq . By the induction hypothesis, Ya,α,q = ConvHull(Ya,α , Ya,βq ) is a normed space with strictly convex norm. Consider the set Xa,α,q = {z ∈ X | cz ∈ Ya,α,q } of all points in X belonging to lines in Ya,α,q . At the same time, we treat such points z as vectors z. The vector p is assumed to be zero. Note that Xa,α,q is the convex hull of the union α ∪ pq. Indeed, Xa,α,q is a convex subset in X containing α ∪ pq: if y, z ∈ Xa,α,q , then either cy = cz or the lines cy and cz bound a normed strip S in X, which determines the segment [cy cz ] ⊂ Ya,α,q . In the former case, the segment [yz] is contained in cy , and in the latter, in S. Therefore, [yz] ⊂ Xa,α,q . On the other hand, if Z ⊂ X is a convex set containing α ∪ pq and z ∈ Xa,α,q , then, representing cz ∈ Ya,α,q as a point in [cx cy ], where x ∈ α and y ∈ pq, we choose a point x ∈ cx so that z ∈ [x y]. Appropriate points cx and cy in the affine space Ya,α,q exist because the linear space Ya,α,q is the direct sum of the subspaces Ya,α and Ya,βq . It follows from z ∈ [x y] that Xa,α,q ⊂ Z. Therefore, Xa,α,q = ConvHull(α ∪ pq). Thus, in fact, Xa,α,q does not depend on the line a: if a ⊂ α is any other line containing p, then Xa ,α,q = ConvHull(α ∪ pq) = Xa,α,q .
(3.4)
Let us define linear operations on vectors in Xa,α,q . The product of a vector z and a number λ is defined by λ · z = w. Here w is the point of the line pz determined by the conditions that d(p, w) = |λ| · d(p, z) and the point z and w are on the same side of p if λ > 0 and on opposite sides of p if λ < 0. The properties (1) 1 · x = x, (2) λ · (μ · x) = (λμ) · x MATHEMATICAL NOTES
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of multiplication follow from the definition. Next, we define the half-sum of vectors u and v by u + v = m, 2 where m is the midpoint of the segment [uv], and the sum u + v by u + v = 2 ·
u + v . 2
Again by definition, we have (3) u + v = v + u, (4) x + p = x, (5) (λ + μ) · x = λ · x + μ · x, (6) x + (−1) · x = p , (7) λ · ( u + v ) = λ · u + λ · v . Property (7) for noncollinear vectors u and v follows from the fact that the convex hull of pu ∪ pv is a two-dimensional normed plane in X. Let us show that Xa,α,q with the operations defined above is a linear space. Properties (1)–(7) give a part of the required conditions. We must also prove the associativity of addition. be the midpoint of the Suppose given three point-vectors x = x, y = y, z = z ∈ Xa,α,q . Let m = m segment [xy], and let k1 = k1 be the point of [zm] dividing this segment in the ratio of 2 : 1. Since the vectors m, z, and k1 belong to the same two-dimensional subspace (see Corollary 5), it follows that ( x + y ) + z = 2 · m + z = 3 · k1 . Therefore, passing to the space Ya,α,q , we obtain ( cx + cy ) + cz = 3 ck1 . Note that, in the affine space Ya,α,q , ck1 is the intersection point of the medians of the triangle with vertices cx , cy , and cz . Similarly, x + ( y + z) = 3 · k2 ,
cx + ( cy + cz ) = 3 ck2 ,
where k2 is the point dividing [xr] in the ratio of 2 : 1 and r is the midpoint of [yz]. Since ck2 is the intersection point of the medians of the triangle with the same vertices cx , cy , and cz in Ya,α,q , it follows that ck1 = ck2 . This means that either k1 = k2 or k1 = k2 and the line k1 k2 is parallel to a. In the latter case, applying (3.4) and an argument similar to that used above to the line a and the space Ya ,α,q , we see that k1 k2 is parallel to any line a contained in α. This contradiction implies that k1 = k2 and (8) ( x + y) + z = x + ( y + z). Properties (1)–(8) show that Xa,α,q is a linear space. The distance d is preserved by translation in Xa,α,q : if v − u = v − u , then d(u, v) = d(u , v ). Indeed, the lines uu and vv are parallel in Xa,α,q and, hence, in X. The case where uu and vv coincide is trivial. If the lines uu and vv are different, then they bound a normed strip in X. The distances d(u, v) and d(u , v ) equal the norms of the corresponding vectors in this strip and, therefore, are equal. A standard verification shows that the function z α,q = d(p, z) is a strictly convex norm on Xa,α,q . The metric d on Xa,α,q is generated by this norm. Now Theorem 1 follows from Proposition 1 and the finite compactness of the space X. MATHEMATICAL NOTES
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4. CONCLUSIONS Theorem 1 proved above makes it possible to shorten the proof of the main result in [4]. In fact, the final proof consists in comparing Theorem 2 (Theorem 4 in [4]) with Theorem 1 of this paper. Importantly, we do not use limits along nonprincipal ultrafilters, which were employed in [4]. This means, in particular, that the truth of the theorems mentioned above does not depend on the axiom of choice. ACKNOWLEDGMENTS The author sincerely thanks the referee for useful comments on the text of the manuscript. This work was supported by the Russian Foundation for Basic Research under grant 14-01-00219 A. REFERENCES 1. H. Busemann, Metric Methods in Finsler Spaces and in the Foundations of Geometry, in Ann. of Math. Stud. (Princeton Univ. Press, Princeton, NJ, 1942), Vol. 8. 2. H. Busemann, “On spaces in which two points determine a geodesic,” Trans. Amer. Math. Soc. 54, 171–184 (1943). 3. H. Busemann, The Geometry of Geodesics (Academic, New York, 1955; Fizmatlit, Moscow, 1962). 4. P. D. Andreev, “Proof of the Busemann conjecture for G-spaces of nonpositive curvature,” Algebra Anal. 26 (2), 1–20 (2014) [St. Petersbg. Math. J. 26 (2), 193–206 (2015)]. 5. H. Busemann, “Spaces with non-positive curvature,” Acta Math. 48, 259–310 (1947). 6. W. Rinow, Die innere Geometrie der metrischen Raume, ¨ in Grundlehren Math. Wiss. (Springer-Verlag, Berlin, 1961), Vol. 105. 7. B. H. Bowditch, “Minkowskian subspaces of non-positively curved metric spaces,” Bull. London Math. Soc. 27 (6), 575–584 (1995). 8. P. D. Andreev, “A. D. Alexandrov’s problem for non-positively curved spaces in the sense of Busemann,” Izv. Vyssh. Uchebn. Zaved. Mat., No. 9, 10–35 (2010) [Russian Math. (Iz. VUZ) 54 (9), 7–29 (2010)]. 9. Ph. K. Hotchkiss, “The boundary of Busemann space,” Proc. Amer. Math. Soc. 125 (7), 1903–1912 (1997).
MATHEMATICAL NOTES
Vol. 101 No. 2 2017
