Lobachevskii Journal of Mathematics Vol. 7, 2000, 43 – 45 \begin{matrix}c \\ ◯ \end{matrix}E.N.Sosov

E.N.Sosov

ON EXISTENCE AND UNIQUENESS OF CHEBYSHEV CENTER OF BOUNDED SET IN A SPECIAL GEODESIC SPACE

(submitted by B.N.Shapukov)

Abstract. In this paper we obtain sufficient conditions for existence and uniqueness of Chebyshev center of nonempty bounded set in a special geodesic space. Analogous conditions for Banach space are well known [1]

1. Preliminary

Let us recall some definitions. A complete metric space (X, ρ) is called a geodesic space [2], if for each pair of points x, y in the space X there exists a point z ∈ X such that

ρ(x, z) = ρ(y, z) = ρ(x, y)/2.

The Chebyshev radius of a nonempty bounded set M in a metric space X [1] is

R(M) = inf_{∈X} sup_{x∈M} ρ(x, y),

and a Chebyshev center of this set is a point z ∈ X such that

sup_{x∈M} ρ(x, z) = R(M).

Henceforth we suppose that the metric space X satisfies some of the following conditions. A. For each pair of points x, y in X there exists a unique point ω(x, y) ∈ X such that

ρ(x, ω(x, y)) = ρ(y, ω(x, y)) = ρ(x, y)/2.

B_{1}. For all p, x, y in X

2ρ(p, ω(x, y)) ≤ ρ(p, x) + ρ(p, y).

Supported by the Russian Fund of Basic Research, Grant N 00 - 01 - 00308.

B_{2}. For all p, x, y in a metric space X

2ρ(ω(p, x), ω(p, y)) ≤ ρ(x, y).

C_{1}. For each r > 0, and for any bounded sequences (p_{n}), (x_{n}), (y_{n}) in X such that for each positive integer n

ρ(p_{n}, x_{n}) ≤ r, ρ(p_{n}, y_{n}) ≤ r, lim_{n→∞} ρ(p_{n}, ω(x_{n}, y_{n})) = r

there holds lim_{n→∞} ρ(x_{n}, y_{n}) = 0. C_{2}. For each r > 0, for each sequence of nonnegative real numbers (ε_{n}) which converges to zero, and for any bounded sequences (p_{n}), (x_{n}), (y_{n}) in a metric space X such that for each positive integer n

ρ(p_{n}, x_{n}) ≤ r + ε_{n}, ρ(p_{n}, y_{n}) ≤ r + ε_{n}, lim_{n→∞} ρ(p_{n}, ω(x_{n}, y_{n})) = r,

there holds the equality lim_{n→∞} ρ(x_{n}, y_{n}) = 0.

Note that the condition (B_{1}) is a simple corollary of (B_{2}), that is a global condition of nonpositivity of a curvature of a metric space according to Buzemann ([3], c. 304). Uniformly convex Banach spaces and Lobachevskii spaces (infinite-dimensional included) are simple examples of complete metric spaces where conditions (A), (B_{2}), (C_{2}) hold true. Now we shall formulate our results.

Theorem 1. If a metric space X satisfies (A), (B_{1}), (C_{1}), then each nonempty bounded set has not more than one Chebyshev center.

Theorem 2. If a complete metric space X satisfies (A), (B_{1}), (C_{2}), then each nonempty bounded set has a unique Chebyshev center.

Theorem 3. If a complete metric space X satisfies (A), (B_{2}), (C_{1}), then each nonempty bounded set has a unique Chebyshev center.

2. Proofs

Proof of the theorem 1. Assume the contrary. Let x, y be two distinct Chebyshev centers of a nonempty bounded set M ⊂ X. Then, from the definition of the Chebyshev center and the conditions (A), (B_{1}), there follow inequalities

sup_{u∈M} ρ(ω(x, y), u) ≤ sup_{u∈M} (ρ(x, u) + ρ(y, u))/2 ≤ R(M).

Therefore ω(x, y) is a Chebyshev center of the set M. From the definition of Chebyshev center it follows that in the set M there exists a sequence (p_{n}) such that

lim_{n→∞} ρ(p_{n}, ω(x, y)) = R(M), ρ(p_{n}, x) ≤ R(M), ρ(p_{n}, y) ≤ R(M) for each n.

By (C_{1}), we get x = y, a contradiction.

Proof of the theorem 2. For a nonempty bounded set M ⊂ X let us consider the family of nonempty closed sets

K_{ε}(M) = {y ∈ X : sup_{x∈M} ρ(x, y) ≤ R(M) + ε}, where ε > 0.

This family has the following two elementary properties :

a) if α ≤ ε then K_{α}(M) ⊂ K_{ε}(M) ;

b) if for a set M there exists a Chebyshev center, then ⋂{K_{ε}(M) : ε > 0} is the set of all Chebyshev centers of M.

ON EXISTENCE AND UNIQUENESS OF CHEBYSHEV CENTER 45

According to Theorem 1 there does not exist more than one Chebyshev center of M. Assume that there does not exist a Chebyshev center of M. Then the set ⋂{K_{ε}(M) : ε > 0} is empty. According to the Cantor theorem ([4], p.

399) there exist c > 0 and a sequence of positive real numbers (ε_{n}) such that lim_{n→∞} ε_{n} = 0, diam(K_{ε_{n}}(M)) ≥ 2c for each n. Therefore for each n in the set K_{ε_{n}}(M) there exist points x_{n}, y_{n} such that

ρ(x_{n}, y_{n}) > c.  (1)

By (A), (B_{1}), the definition of Chebyshev radius and of K_{ε}(M) we obtain

R(M) ≤ sup_{z∈M} ρ(z, ω(x_{n}, y_{n}) ≤ sup_{z∈M} (ρ(z, x_{n}) + ρ(z, y_{n}))/2 ≤ R(M) + ε_{n} (2)

for each n. Consequently in M there exists a sequence (z_{n}) such that

lim_{n→∞} ρ(z_{n}, ω(x_{n}, y_{n})) = R(M).  (3)

By (C_{2}) we obtain lim_{n→∞} ρ(x_{n}, y_{n}) = 0, which contradicts inequality (1).

Proof of the theorem 3. The beginning of the proof of this theorem coincides with the proof of the theorem 2 except for its last sentence. Let us consider for each n points u_{n}, v_{n} in geodesic segments [z_{n}, x_{n}], [z_{n}, y_{n}], respectively,y such that

ρ(z_{n}, u_{n}) = λ_{n}ρ(z_{n}, x_{n}), ρ(z_{n}, v_{n}) = λ_{n}ρ(z_{n}, y_{n}), where λ_{n} = R(M)/(R(M)+ ε_{n}). Hence, by (A), (B_{2}) and the inequalities (2), we obtain for each n auxiliraly inequalities: ρ(ω(u_{n}, v_{n}), ω(x_{n}, y_{n})) ≤ ρ(ω(u_{n}, v_{n}), ω(x_{n}, v_{n})) + ρ(ω(x_{n}, v_{n}), ω(x_{n}, y_{n})) ≤ ≤ (ρ(x_{n}, u_{n}) + ρ(y_{n}, v_{n}))/2 = (1 − λ_{n})(ρ(z_{n}, x_{n}) + ρ(z_{n}, y_{n}))/2 ≤ ε_{n}, ρ(z_{n}, ω(x_{n}, y_{n}))−ρ(ω(u_{n}, v_{n}), ω(x_{n}, y_{n})) ≤ ρ(z_{n}, ω(u_{n}, v_{n})) ≤ ρ(z_{n}, ω(x_{n}, y_{n}))+ ρ(ω(u_{n}, v_{n}), ω(x_{n}, y_{n})). From these inequalities and the equality (3) we obtain lim_{n→∞} ρ(z_{n}, ω(u_{n}, v_{n})) = R(M). Therefore, by (C_{1}), lim_{n→∞} ρ(u_{n}, v_{n}) = 0. From this equality and the inequalities ρ(x_{n}, y_{n}) ≤ ρ(x_{n}, u_{n}) + ρ(u_{n}, v_{n}) + ρ(v_{n}, y_{n}) = (1 − λ_{n})(ρ(z_{n}, x_{n}) + ρ(z_{n}, y_{n})) + ρ(u_{n}, v_{n}) ≤ 2ε_{n}+ρ(u_{n}, v_{n}) for each n we get lim_{n→∞} ρ(x_{n}, y_{n}) = 0. This contradicts the inequality (1). Thus the theorem 3 is proved.

References

[1] Garkavi A. L., About a minimizing net and a minimizing section of set in a normed space, Proc. of the Acad. of Sci. USSR, Ser. math., V. 26, No. 1, 87-106 (1962). (Russian)

[2] Efremovich V. A., Euclidean space and Lobachevskii space are not equimorphic, Uspekhi Mat. Nauk, V. 4, No. 2 (30), 178-179, (1949). (Russian)

[3] Buzemann H., Geometry geodesic, Moscow, ”Fizmatgis” Publishing House, 1962. (Russian)

[4] Engelking R., General topology, Moscow, ”Mir” Publishing House, 1986. (Russian)

Kazan Pedagogical University, ul. Mejlauk, 1, Kazan: 420021, Russia E-mail address: ens@kspu.kcn.ru

Received July, 16, 2000