ON THE FILLING RADIUS OF POSITIVELY CURVED ALEXANDROV SPACES

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1 ON THE FILLING RADIUS OF POSITIVELY CURVED ALEXANDROV SPACES TAKUMI YOKOTA Abstract. It was shown by F. Wilhelm that Gromov s filling radius of any positively curved closed Riemannian manifolds are less than that of the round sphere unless they are isometric to each other. In this short paper, we adapt his proof to see that the same is true for any positively curved closed Alexandrov spaces as well. 1. Introduction In this paper, we are concerned with the metric invariant Fill Rad, the filling radius, of closed Riemannian manifolds that was introduced by Gromov [Gr]. Before giving its definition, we first recall that the map i : x d(, x) gives an isometric embedding of any metric space (X, d) of bounded diameter Diam(X) < into the space (L (X), ) of bounded Borel functions on X, called the Kuratowski embedding. Definition 1 (Gromov [Gr, Section 1]). The filling radius Fill Rad(V ) of a closed Riemannian manifold V is the infimum of ε > 0 for which its fundamental homology class [V ] H (V ; A), with the coefficient ring A = Z or Z 2 depending on whether V is oriented or not, is mapped to zero under the inclusion of V into the ε-neighborhood of i(v ) in L (V ). Among a number of works inspired by Gromov s paper [Gr], Wilhelm [Wi] established interesting comparison and rigidity theorems for closed positively curved manifolds stated as follows: Theorem 2 (Wilhelm [Wi]). For any n-dimensional closed Riemannian manifold V of sectional curvature 1, either Fill Rad(V ) < Fill Rad(S n ) or V is isometric to the round sphere S n of constant curvature 1. Theorem 2 can be applied to any positively curved closed manifolds by the rescale of the metrics. The proof of Theorem 2 in [Wi] relies on Lemma 4 below shown by Katz [Ka]. Definition 3 ([Wi]). For any metric space X = (X, d), we define its spread, denoted Spread(X), as the infimum of R > 0 for which there is a subset Y X of Diam(Y ) R such that d(x, Y ) R for any x X Mathematics Subject Classification. 53C23. Key words and phrases. Alexandrov space, filling radius, packing radius. This work was partially supported by Grant-in-Aid for Research Activity (startup) No

2 2 T. YOKOTA Lemma 4 (Katz [Ka]). For any closed Riemannian manifold V, Fill Rad(V ) Spread(V ). 1 2 Katz [Ka] used Lemma 4 to calculate the filling radius of the round sphere Fill Rad(S n ) = 1 2 Spread(Sn ) = l n /2. Here and throughout this paper, we use l n := arccos( 1 n+1 ). Actually, l n is the spherical distance between vertices of a regular (n+1)-simplex whose vertices lie on the unit sphere S n R n+1, and the set Y := {p 1,..., p n+2 } S n of its vertices gives Spread(S n ) = Diam(Y ) = l n. Lemma 4 reduces the estimate of the filling radius to that of the spread. In fact, Wilhelm [Wi] proved the following theorem, from which Theorem 2 follows immediately. Theorem 5 (Wilhelm [Wi]). For any n-dimensional closed Riemannian manifold V of sectional curvature 1, either Spread(V ) < Spread(S n ) = l n or V is isometric to the round sphere S n. An easy estimate in [Ka] says that Spread(X) 2 3 Rad(X) 2 3 Diam(X) for any length space X. Here, Rad(X) := inf x X sup y X d(x, y) stands for the radius of a metric space (X, d). For the real projective space, these inequalities become equalities [Ka]. Now the Bonnet Myers diameter bound yields that Spread(V ) 2 3π for any Riemannian manifolds of sectional curvature 1, however Theorem 5 is sharper than this estimate. We also remark that, in spite of Theorem 2, the following question by Gromov still remains open. Question 6 (Gromov [Gr, p. 130]). Is it true that, for any n-dimensional closed Riemannian manifold V of scalar curvature σ 2 > 0, Fill Rad(V ) Const n σ 1? The goal of this paper is to extend Theorem 5 to finite-dimensional Alexandrov spaces of curvature 1, defined in Definition 14 below. By dimension, we mean the covering dimension or the Hausdorff dimension, which are known to coincide for any Alexandrov space with a lower curvature bound. Here we state the main theorem of the present paper. Theorem 7. For any n-dimensional Alexandrov space X of curvature 1, either Spread(X) < Spread(S n ) = l n or X is isometric to the round sphere S n. In order to state corollaries of our Theorem 7, we recall some facts. At first, any finite-dimensional Alexandrov space X with a lower curvature bound admits a fundamental homology class and hence its filling radius is defined, provided it has empty boundary X = ; see Yamaguchi [Ya], Grove Petersen [GP]. Secondly, as was remarked by Wilhelm [Wi], Theorem 2 remains true with Fill Rad replaced with Cont k Rad for each 0 k n 1, because Lemma 4 remains true after such replacement. Although we do not give its definition, we just recall that Fill Rad(V ) Cont n 1 Rad(V ) Cont 0 Rad(V ) 1 2 Spread(V ) for any closed Riemannian manifold V ; see [Gr, Appendix 2]. Now we have Corollary 8. For any n-dimensional Alexandrov space X of curvature 1 with X =, either Fill Rad(X) < Fill Rad(S n ) or X is isometric to the round

3 ON THE FILLING RADIUS OF POSITIVELY CURVED ALEXANDROV SPACES 3 sphere S n. Moreover, for any n-dimensional Alexandrov space X of curvature 1, either Cont k Rad(X) < Cont k Rad(S n ) = l n /2 for any 0 k n 1 or X is isometric to the round sphere S n. Several remarks on Theorem 7 are in order here. Remark 9. To be precise, Theorem 5 is stated in [Wi] for closed Riemannian manifolds of sectional curvature 1 and their Gromov Hausdorff limits. Recall that they are typical examples of Alexandrov spaces of curvature 1, and Spread( ) is continuous under the Gromov Hausdorff convergence of compact metric spaces. However, we know that there exist Alexandrov spaces of curvature 1 which cannot be obtained as Gromov Hausdorff limits of any sequence of closed Riemannian manifolds with the same lower curvature bound; see [PWZ]. Therefore, our Theorem 7 is a natural generalization of Theorem 5. Remark 10. It seems interesting to know whether Theorem 7 has its infinitedimensional analogue. Namely, we may ask whether it is true that Spread(X) π/2 for any infinite-dimensional Alexandrov space X of curvature 1. As an example, the unit sphere S of any infinite-dimensional Hilbert space equipped with the angle metric has Spread(S ) = π/2. Indeed, by considering its orthonormal basis, it is easy to see that Spread(S ) π/2. Moreover, if Y S is a subset of Diam(Y ) < π/2, then the antipode y of any y Y satisfies ( y, Y ) > π/2. This means Spread(S ) π/2. Apart from (S, ), there seem to be many infinite-dimensional Alexandrov spaces of curvature 1 with spread = π/2. In Theorem 7, we assume the finite-dimensionality of X to guarantee the compactness of X itself as well as of the space of directions Σ x at any point x X. By examining its proof, we know that Spread(X) π/2 for any infinitedimensional Alexandrov space X of curvature 1 whose space of directions Σ x, at any x X, is a compact Alexandrov space of curvature 1. However, the author does not know whether there exists such an infinite-dimensional Alexandrov space. A result in this direction was obtained by Berestovskii Plaut; see [BP, Theorem 1.6]. Remark 11. If any Alexandrov space X of curvature 1 admits a surjective 1-Lipschitz map of the round sphere onto X, it would yield a short proof of Theorem 7. In [Pet], Petrunin introduced a map gexp p (1; ) called the gradient exponent map, which could be useful for our purpose, for each p X of any Alexandrov space X of curvature 1. However, the domain of this map is not all of the spherical suspension Σ(Σ p ) of the space of directions Σ p, and we have no idea how to make use of this map to prove Theorem 7. We thank A. Mitsuishi for informing us of this. After we prepare the notations and recall relevant facts in Section 2, we present a proof of Theorem 7 in Section 3. We largely follow Wilhelm s argument in [Wi], however we have to modify it because of some techniques used there that cannot be available for general Alexandrov spaces. Moreover, as the proof of Theorem 5 given in [Wi] looks somewhat involved, a simplification of the proof is also our contribution.

4 4 T. YOKOTA For further information on Gromov s filling invariants, the reader is encouraged to consult his seminal paper [Gr]. 2. Preliminary In this section, we prepare some terminologies and facts that will be needed later. We sometimes use some of them without giving their definitions if they are standard and common in Alexandrov geometry. For the precise definitions and basics of the theory of Alexandrov spaces, we refer the reader to the fundamental paper [BGP] by Burago Gromov Perelman, or [BBI, Chapter 10], or [Pl]. We start with the definition of Alexandrov spaces with lower curvature bounds. We first recall that, for a fixed real number κ R, κ (x; y, z) [0, π] denotes the comparison angle for any triple (x, y, z) consisting of distinct three points of a metric space (X, d). For example, if κ = 1, which is the case of our interest, it is defined by cos 1 (x; y, z) := cos(d(y, z)) cos(d(x, y)) cos(d(x, z)) sin(d(x, y)) sin(d(x, z)) for any triple (x, y, z) in X whose perimeter peri(x, y, z) := d(x, y) + d(y, z) + d(z, x) is less than 2π. Definition 12. Let κ R. We say that a metric space (X, d) has curvature κ, if the following holds: any quadruple (x; y, z, w) consisting of distinct four points of X of size, i.e., the maximum of the perimeters of all triples in it, less than 2π/ κ if κ > 0, satisfies that (13) κ (x; y, z) + κ (x; z, w) + κ (x; w, y) 2π. Condition (13) is sometimes called the quadruple condition. Definition 14. Let κ R. We say that a metric space (X, d), which is not necessarily a geodesic space, is an Alexandrov space of curvature κ if it satisfies the following axioms: (1) (X, d) is a complete length space, i.e., the distance between any two points is given by the infimum of the lengths of curves joining them, and (2) (X, d) has curvature κ in the sense of Definition 12. We say that (X, d) is an Alexandrov space when it is an Alexandrov space of curvature κ for some κ R. An usual convention is that a one-point set and a two-point set equipped with the distance of diameter = π are 0-dimensional Alexandrov spaces of curvature 1. Any 1-dimensional Alexandrov space of curvature κ is a 1-dimensional Riemannian manifold possibly with boundary. Next, we define the inner product in general metric spaces, which is similar to the one introduced by Berg Nikolaef [BN] in their search for new characterizations of CAT(0) spaces. For any triple (x, y, z) in a metric space (X, d) and κ R, we put xy, xz κ := d(x, y)d(x, z) cos κ (x; y, z). We define xy, xz κ := 0 if d(x, y) d(x, z) = 0. When κ > 0, d(x, y) d(x, z) > 0, and κ (x; y, z) is not well-defined, we declare that xy, xz κ := +.

5 ON THE FILLING RADIUS OF POSITIVELY CURVED ALEXANDROV SPACES 5 Here we collect fundamental properties of Alexandrov spaces. In the statement below and thereafter, we use the notation (a, b) := {x X \ {a, b} d(a, x) + d(x, b) = d(a, b)} for any points a, b of a metric space (X, d), and M n κ denotes the n-dimensional model space, i.e., the simply-connected complete Riemannian manifold of constant curvature κ. Proposition 15. Let (X, d) be an Alexandrov space of curvature κ. Then the following hold. (16) (1) (Triangle comparison) For any triple (x, y, z) in X, of peri(x, y, z) < 2π/ κ if κ > 0, there is its isometric copy ( x, ỹ, z) in the model surface M 2 κ, and for any point w (y, z) and the corresponding point w (ỹ, z) with d(ỹ, w) = d(y, w), d(x, w) d( x, w). (2) (Angle comparison) For any points x, y, z X and ẑ (x, z), κ (x; y, ẑ) κ (x; y, z). (3) (Lang Schroeder Sturm inequality [St], cf. [Yo]) For any p X and finite sequences {x i } X I of points and (λ i ) i I (R + ) I of positive real numbers, with I := {1,..., N}, i,j I λ i λ j px i, px j κ 0. Due to the angle comparison above, the angle (γ, η) := lim κ (p; γ(s), η(t)) = sup κ (p; γ(s), η(t)) s,t 0 s,t>0 is well-defined for any two geodesics γ, η : [0, δ) X with p := γ(0) = η(0). The angle does not depend on the lower curvature bound κ R. The space of directions (Σ x, ) at a point x X is the metric completion of the set (Σ x, ) of equivalence classes of unit speed geodesics departing from x. The tangent cone C x at x is defined as the Euclidean cone over Σ x. Following Petrunin [Pet], we use the notations y x Σ x and y x y x, respectively, to denote the set of all equivalence classes of geodesics from x to y X and its element. Finite-dimensional Alexandrov spaces are known to be locally compact [BBI, Corollary ], and hence they are geodesic spaces, i.e., there is a (minimal) geodesic realizing the distance between any two points. For geodesic spaces, each of conditions (1) (3) in Proposition 15 are equivalent to the quadruple condition (13). Remark 17. Inequality (16) was at first proved by Lang Schroeder in the appendix of [LS] in tangent cones of Alexandrov spaces with p being its base point. Subsequently, Sturm [St] obtained an inequality equivalent to (16) for Alexandrov spaces by using the result of [LS] and the angle comparison. In [Yo], we formulated the Lang Schroeder Sturm inequality as in (16). We established the following rigidity theorem in [Yo].

6 6 T. YOKOTA Theorem 18 ([Yo, Theorem B]). Let (X, d) be an Alexandrov space with curvature κ. Suppose that we have a point p := x 0 X and finite sequences (x i ) i I X I of points and (λ i ) i I (R + ) I, with I := {1,..., N}, such that λ i λ j px i, px j κ = 0. i,j I Then Y := {x i i = 0, 1,..., N} is isometrically embeddable into the (N 1)- dimensional model space Mκ N 1, and conv Y is isometric to the closed convex hull of the embedded image in M N 1 κ. The definition of the closed convex hull conv Y of a subset Y X, which appears in the above statement, is given in the original paper [Yo]. Next we consider the packing radius of positively curved Alexandrov spaces. Definition 19. Let (X, d) be a metric space and q 2 be an integer. define its q-th packing radius pack q (X) by (20) pack q (X) := 1 { } 2 sup min d(x i, x j ) (x i ) q 1 i<j q i=1 Xq. The sequence (x i ) X q is called a q-th packer when it attains the supremum in (20). We have the following comparison and rigidity results for packing radius of positively curved Alexandrov spaces. Proposition 21 (Grove Wilhelm [GW], cf. [Yo]). Let (X, d) be an Alexandrov space of curvature 1 and q 2. Then (1) pack q (X) pack q (S n ) = l q 2 /2 for n q 2. (2) If pack q (X) = pack q (S q 2 ) and there exists a q-th packer, then X is isometric to the spherical join S q 2 Y for some Alexandrov space Y of curvature 1. Proof. This proposition was established in [GW] for finite-dimensional Alexandrov spaces. Now inequality (16) and Theorem 18 yield an immediate proof for possibly infinite-dimensional Alexandrov spaces. Part (1) follows easily from the Lang Schroeder Sturm inequality (16). Indeed, if there exist points (x i ) q i=1 in X such that d(x i, x j ) l q 2 for each i j, then 1 (x i ; x j, x k ) l q 1 for any i j k, cf. [Wi, Proposition A.2]. This contradicts inequality (16) unless d(x i, x j ) = l q 2 for each i j. For Part (2), if we have a q-th packer (x i ) X q giving pack q (X) = pack q (S q 2 ), the above observation implies that d(x i, x j ) = l q 2 and 1 (x i ; x j, x k ) = l q 1 for any i j k, and Theorem 18 yields that x i s are contained in the subset isometric to the round sphere S q 2. Then we appeal to the maximum diameter theorem, e.g. [Mi], to finish the proof. Shteingold [Sh] studies other metric invariants, the covering radius and the paving diameter, defined in similar fashion. We will prove, in Lemma 25, a relation between the packing radius and the spread of any Alexandrov space of curvature 1. In its proof, we will make frequent use of the following lemma We

7 ON THE FILLING RADIUS OF POSITIVELY CURVED ALEXANDROV SPACES 7 with or without mentioning it; for its proof, see e.g. [BBI, Corollary 4.5.7], cf. [Pl, Proposition 49]. Proposition 22 (First variation formula). Suppose that x, y, and p are points of a locally compact Alexandrov space (X, d), and a geodesic segment xy representing y x Σ x is given. Then, for any z xy sufficiently close to x, we have d(p, z) = d(p, x) d(x, z) cos pxy + o(d(x, z)) as z x, where we used pxy := ( y x, p x) = inf { ( y x, p x) p x p x}. Finally, we give a definition of regular points of certain functions and its relevant properties. These are nothing but those for more general functions called admissible by Perelman in [Per, Per2], which play crucial role in the proof of his celebrated stability theorem for Alexandrov spaces. For details, we refer to Perelman s original papers [Per, Per2], or Kapovitch s survey [Kap]. Definition 23. Let ε > 0 and f : U R k be a function defined on an open set U of a finite-dimensional Alexandrov space (X, d) of curvature κ, and suppose that each coordinate function of f = (f 1,..., f k ) is given by the distance function f i ( ) = d(, p i ) from a point p i X. Following Perelman [Per, Per2], we say that f is ε-regular at x U if κ (x; p i, p j ) > π/2 + ε for any i j, and there is a point p 0 X such that κ (x; p i, p 0 ) > π/2 + ε for any i. We say that f is regular at x U if it is ε-regular at x U for some ε > 0. We collect some facts about regular points, e.g. [Per2, Lemma 2.3 (2)], cf. [Kap, Lemma 6.7]. Lemma 24. Let f : U R k be regular at x X. Then (1) k n; (2) The set of regular (resp. ε-regular) points of f is open; (3) f : U R k is 1-Lipschitz on U with respect to the norm x := max 1 i k x i on R k ; (4) f is co-lipschitz around x, i.e., for any small R > 0, f(b(x, R)) B(f(x), cr) for some small constant c > 0, where B(x, ) is the open metric ball centered at x. In the proof of Theorem 5, Wilhelm [Wi] used smooth approximation of the distance functions. However, the above properties of regular points will turn out to be enough for our purpose. 3. Proof of the Main theorem In this section, we describe the proof of Theorem 7. To begin with, we state the main lemma. This is essentially established in [Wi] for closed Riemannian manifolds of sectional curvature 1, although it is not stated explicitly there. Theorem 7 follows immediately from Lemma 25 below and Proposition 21. Lemma 25 (cf. [Wi, Main Lemma 8]). For any n-dimensional Alexandrov space X of curvature 1, (26) Spread(X) max { 2 pack n+2 (X), π/2 } l n.

8 8 T. YOKOTA In Lemma 25, we use pack n+2 (X) to bound Spread(X), because pack q (X) pack q 1 (X), and pack n+3 (X) π/2 for any n-dimensional Alexandrov space X of curvature 1, cf. Lemma 24. (1). The second inequality in (26) is a consequence of Proposition 21. Before beginning the proof of Lemma 25, we explain how Theorem 7 follows from it. Obviously, Lemma 25 yields a part of Theorem 7. If Spread(X) = l n, then Spread(X) = 2 pack n+2 (X) = l n. Since any finite-dimensional Alexandrov spaces of curvature 1 is compact, there exist points (x i ) X n+2 such that d(x i, x j ) = l n for any i j. Then the rigidity part of Theorem 7 follows from that of Proposition 21. Proof of Lemma 25. First of all, we may assume that n 2, because the case n 1 is trivial. We fix any λ > max { 2 pack n+2 (X), π/2 }, and take a maximal subset Y := {p 1,..., p k } of X such that d(p i, p j ) = λ for any i j. Note that λ > π/2, and we let λ > π/2 denote the angle at the vertices of the regular triangle of side length λ in S 2, as in [Wi]. We put Y := {p 1,..., p k 1 }. Next, we let I = I k 1 := {x X d(x, p i ) = λ for any p i Y }, and take a maximal subset R := {r 0,..., r s } of I containing r 0 := p k I such that d(r i, r j ) > λ for any i j. Note that Y R = k + s < n + 2. Now, we prove the following Lemma 27 (cf. [Wi, Lemma 10]). Y R is a λ-net in X, by which we mean that d(x, Y R) < λ for any x X. Proof. We prove this by contradiction. We suppose that Y R is not a λ-net, and let z X be the point such that d(z, Y R) = max x X d(x, Y R) λ. It follows from the following well-known fact that such z is uniquely determined. Fact 28 (e.g. [GW, Lemma 2.1]). Let (X, d) be an Alexandrov space of curvature 1, and suppose that a subset Y X satisfies sup x X d(x, Y ) > π/2. Then there exists a unique point z := A(Y ) X such that d(z, Y ) = max x X d(x, Y ). In particular, the antipodal map A : { x X sup y X d(x, y) > π/2 } X defined by A(x) := A({x}) is well-defined. Moreover, for any x X and Y X, either d(x, Y ) π/2 or d(x, Y ) + d(x, A(Y )) π. We will apply this fact not only to (X, d) but also to the space of directions (Σ ξx, ξx ) at a vector ξ x Σ x for some x X. At first, we verify the following claims. Claim 29 (cf. [Wi, Lemma 18]). d(z, p i ) = d(z, p j ) for any p i, p j Y Proof. Suppose that d(z, p i0 ) > d(z, Y ) for some p i0 Y. Then any point ẑ (z, p i0 ) close to z satisfies that d(ẑ, p i0 ) > d(z, Y ) and, by 1 (z; p i0, p i ) > π/2 and the triangle comparison, d(ẑ, p i ) > d(z, p i ) d(z, Y R) for any p i Y R\{p i0 }. This contradicts the choice of z and proves that d(z, p i ) = d(z, Y ) for any p i Y. We put C := {x X d(x, Y R) λ}. Claim 30 (cf. [Wi, Lemma 19]). For any x C, there is a vector ξ x Σ x such that (1) (ξ x, p 1 x ) π λ < π/2;

9 ON THE FILLING RADIUS OF POSITIVELY CURVED ALEXANDROV SPACES 9 (2) (ξ x, p i x ) = (ξ x, p j x ) for each p i, p j Y ; (3) (ξ x, r i x ) > π/2 for each r i R. Proof. For each x C, we define a function u(ξ) := max { (ξ, p i x ) p i Y } on Σ x, and take ξ x Σ x such that u(ξ x ) = min {u(ξ) ξ Σ x }. Since the space of directions (Σ x, ) is (n 1)-dimensional Alexandrov space of curvature 1 and hence is compact, e.g. [BBI, Corollary ], such ξ x always exists. Now we confirm that ξ x satisfies the desired properties. In the proof of this claim, we use the classical notation (η) as in [BGP] etc, instead of η ξ Σ ξ, for vectors ξ, η Σ x. In addition, A = A denotes the antipodal map, recalled in Fact 28, of the space of directions Σ at = x or ξ x. Since ( p i x, p j x ) 1 (x; p i, p j ) λ > π/2 for each i j, by Fact 28, A x ( p k x ) Σ x satisfies that u(ξ x ) u(a x ( p k x )) = max (A x ( p k x ), p i x ) π λ. p i Y This proves Part (1). For each p i Y, we fix p i x Part (2) is not true, then p i x satisfying that (ξ x, p i x ) = (ξ x, p i x ). If (ξ x, p i 0 x ) = (ξ x, p i 0 x ) < max (ξ x, p i x ) = u(ξ x ) p i Y for some p i0 Y. Since ξx (( p i 0 x ) ), ( p i x ) ) 1 (ξ x ; p i 0 x, p i x ) > π/2, by Fact 28, A ξx (( p i 0 x ) ) satisfies that ξx (A ξx (( p i 0 x ) ), ( p i x ) ) < π/2 for each i i 0. Therefore, some vector ξ Σ x close to ξ x with a vector in (ξ) Σ ξx close to A ξx (( p i 0 x ) ) satisfies that (ξ, p i 0 x ) < u(ξ x ) and (ξ, p i x ) < (ξ x, p i x ) u(ξ x ) for each i i 0. This contradicts the choice of ξ x. To see Part (3), we assume that (ξ x, r x) = (ξ x, r x) π/2 for some r R and r x r x. Then, since ( p i x, r x) 1 (x; p i, r) > π/2 for any p i Y, this implies ξx (( p i x ), ( r x) ) 1 (ξ x ; p i x, r x) > π/2, and, by Fact 28, A ξx (( r x) ) satisfies that ξx (A ξx (( r x) ), ( p i x ) ) < π/2 for any p i Y. Therefore, some vector ξ Σ x close to ξ x with a vector in (ξ) Σ ξx close to A ξx (( r x) ) satisfies that (ξ, p i x ) < (ξ x, p i x ) for each p i Y. This contradicts the choice of ξ x. Now, for any fixed δ > 0, we construct a sequence z 0 := z, z 1,..., z N of points of X with L(N) := N l=1 d(z l 1, z l ) < such that λ d(z N, p 1 ) < λ + δ, and for each l = 1,..., N, d(z l, p i ) d(z l, p j ) < δ L(l)/L(N) for each p i, p j Y ; d(z l, p 1 ) < d(z l 1, p 1 ), and d(z l, r i ) > d(z l 1, r i ) for any r i R. This is done as follows: for given z l, we take ẑ l+1 X with ẑl+1 z l Σ zl close to ξ zl chosen in Claim 30, and choose z l+1 (z l, ẑ l+1 ) close to z l. Then, Proposition 22 yields the desired properties. In case the sequence {z l } accumulates to some point z X with d(z, p 1 ) λ + δ, we can restart from that point. Letting δ 0, we see that z N = zn(δ) δ subconverges to a point r X such that r I and d( r, r i ) > λ for any r i R, which contradicts the maximality of R. Therefore it completes the proof of Lemma 27. We continue the proof of the main Lemma 25.

10 10 T. YOKOTA Lemma 31 (cf. [Wi, Lemma 9]). R consists of a single point p k, i.e., Y R = Y. Proof. Suppose that there is a point r R I satisfying d(r, p k ) > λ, and put Y + := {p 1,..., p k, r}. Notice that Y + = k + 1 < n + 2. Now, a map f : X R k 1 defined by f(x) := (d(x, p 1 ),..., d(x, p k 1 )) is regular at r R. Since f is co-lipschitz on arbitrary small neighborhoods U r of r and U pk of p k by Lemma 24, we can find ˆp 0,1 ˆp 0,2 U r and ˆp k U pk with ˆp 0,1, ˆp 0,2 f 1 (f(ˆp k )). Indeed, if this is not true, f is injective on U r and hence it is bi-lipschitz to an open set f(u r ) of R k 1. This is a contradiction because U r contains a bi-lipschitz copy of an open subset of R n, e.g. [BBI, Theorem ]. We may assume that d(ˆp 0,1, ˆp k ) d(ˆp 0,2, ˆp k ), and put ˆp 0 := ˆp 0,1 and Ŷ := {p 1,..., p k 1, ˆp k, ˆp 0 }. We fix ε > 0 so small that ˆp 0,2 f 1 (f(ˆp k )) is contained in the set Ŷ ε reg := { x X 1 (x; p, p ) > π/2 + ε for any p p Ŷ We notice that Ŷ reg ε contains the set { } x f 1 (I(f(ˆp k ), δ)) d(x, ˆp k ), d(x, ˆp 0 ) > ψ(δ) for some δ = δ(ε) > 0 and ψ(δ) > 0 with ψ(δ) 0 as ε, δ 0. Here, { } I(f(ˆp k ), δ) := (v i ) R k 1 vi d(ˆp k, p i ) < δ for any 1 i k 1. Using the following proposition, for each x Ŷ reg, ε we find w x Σ x such that (w x, ˆp 0 x ) > π/2 + ε, (w x, ˆp k x ) < π/2 ε, and (w x, p i x ) = π/2 for any p i Y. Proposition 32 (cf. Perelman [Per2, Lemma 2.3]). Let (Σ, ) be a compact Alexandrov space of curvature 1, and suppose that a finite number of subsets B i Σ, i = 0, 1,..., k, satisfy that (B i, B j ) > π/2 + ε for each i j. Then there exists a vector w Σ such that (w, B 0 ) > π/2 + ε, (w, B k ) < π/2 ε, and (w, B i ) = π/2 for any 1 i k 1. Proof. For reader s convenience, we explain how to chose w Σ in Proposition 32; it is chosen as an element w Σ satisfying that (w, B k ) = min { (v, B k ) v Σ 0}, where Σ 0 := {v Σ (v, B i ) π/2 for any 1 i k}. Then, by Fact 28, A(B 0 ) Σ 0 and (w, B k ) (A(B 0 ), B k ) < π/2 ε. Furthermore, it also follows from the choice of w that (w, B i ) = π/2 for any 1 i k 1 and (w, B 0 ) > π/2 + ε. Remark 33. We comment that if p i x = { p i x } consists of a single vector for each 1 i k, and the set { p i x } k i=1 spans a Euclidean space in the tangent 1 cone C x at x X, we can take w x as the orthogonal projection of p k x subspace spanned by { p i x } k 1 }. w w in Σ x, where w C x is Σ x onto the orthogonal complement of the i=1. Due to Proposition 34 below, w = 0, and by x ) < π/2 c(ε) for definition, (w x, p i x ) = π/2 for each p i Y and (w x, p k some c(ε) > 0.

11 ON THE FILLING RADIUS OF POSITIVELY CURVED ALEXANDROV SPACES 11 Proposition 34 ([Wi, Proposition 17]). If {v 1,..., v k, v} is a subset of a Hilbert space satisfying that (v i, v j ) > π/2 and (v i, v) > π/2 for any 1 i j k, then {v 1,..., v k } is linearly independent. We return to the proof of Lemma 31. For any fixed δ (0, δ(ε)), we construct a sequence q 0 := ˆp 0,2, q 1,..., q N in Ŷ reg ε with L(N) := N l=1 d(q l 1, q l ) < such that λ < d(q N, ˆp k ) < λ + δ, and for each l = 1,..., N, d(q l, ˆp k ) < d(q l 1, ˆp k ) and d(q l, ˆp 0 ) > d(q l 1, ˆp 0 ); d(q l, p i ) d(ˆp k, p i ) < δ L(l)/L(N) for each p i Y. This is done as follows: If the point q l is given, we take a point ˆq l+1 X with ˆq l+1 q l Σ ql close to w ql chosen above, and choose q l+1 (q l, ˆq l+1 ) close to q l. Proposition 22 yields the desired properties. In case the sequence {q l } accumulates to some point q X with d(q, ˆp k ) λ + δ, we can restart from that point. Letting δ 0, we see that q N = qn(δ) δ subconverges to a point ˆq X with ˆq f 1 (f(ˆp k )) and d(ˆq, ˆp k ) = λ. Next, Letting ˆp k p k, we see that ˆq subconverges to a point q in I = f 1 (f(p k )) with d(q, p k ) = λ. This contradicts the maximality of Y and completes the proof of Lemma 31. Now we know that Y is a λ-net in X, Diam(Y ) = λ, and hence Spread(X) λ. Since λ > max { 2 pack n+2 (X), π/2 } is chosen arbitrarily, this completes the proof of Lemma 25. Acknowledgements. The author thanks Ayato Mitsuishi for discussions at the early stage of this work. He also thanks the referee for careful reading of the manuscript. References [BN] I.-D. Berg and I.-G. Nikolaev, Quasilinearization and curvature of Aleksandrov spaces. Geom. Dedicata 133 (2008), [BP] V. Berestovskii and C. Plaut, Homogeneous spaces of curvature bounded below. J. Geom. Anal. 9 (1999), no. 2, [BBI] D. Burago, Y. Burago, and S. Ivanov, A course in metric geometry. Graduate Studies [BGP] in Mathematics, 33. American Mathematical Society, Providence, RI, Y. Burago, M. Gromov, and G. Perelman, A. D. Aleksandrov spaces with curvatures bounded below. (Russian) Uspekhi Mat. Nauk 47 (1992), no. 2(284), 3 51, 222; translation in Russian Math. Surveys 47 (1992), no. 2, [Gr] M. Gromov, Filling Riemannian manifolds. J. Diff. Geom. 18 (1983), no. 1, [GP] K. Grove and P. Petersen, A radius sphere theorem. Invent. Math. 112 (1993), no. 3, [GW] K. Grove and F. Wilhelm, Hard and soft packing radius theorems. Ann. of Math. (2) 142 (1995), no. 2, [Kap] V. Kapovitch, Perelman s stability theorem. Surveys in differential geometry. Vol. XI, , Int. Press, Somerville, MA, [Ka] M. Katz, The filling radius of two-point homogeneous spaces. J. Diff. Geom. 18 (1983), [LS] no. 3, U. Lang and V. Schroeder, Kirszbraun s theorem and metric spaces of bounded curvature. Geom. Funct. Anal. 7 (1997), no. 3,

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