c Allerton Press, Inc., 2008.
ISSN 1066-369X, Russian Mathematics (Iz. VUZ), 2008, Vol. 52, No. 4, pp. 59–64. c E.N. Sosov, 2008, published in Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, 2008, No. 4, pp. 66–72.
Original Russian Text The Relative Chebyshev Centers of Finite Sets in Geodesic Spaces
E. N. Sosov1*
1
Kazan State University, ul. Kremlyovskaya 18, Kazan, 420008 Russia
Received May 17, 2007
Abstract—In the present paper we estimate variation in the relative Chebyshev radius RW (M ),
where M and W are nonempty bounded sets of a metric space, as the sets M and W change. We
find the closure and the interior of the set of all N -nets each of which contains its unique relative
Chebyshev center, in the set of all N -nets of a special geodesic space endowed by the Hausdorff
metric. We consider various properties of relative Chebyshev centers of a finite set which lie in this
set.
DOI: 10.3103/S1066369X08040075
Key words and phrases: relative Chebyshev center, Hausdorff metric, geodesic space.
1. PRELIMINARY
Let (X, ρ) be a metric space. We introduce the following notation. R+ is the set of all nonnegative real
numbers. |xy| = ρ(x, y) for x, y ∈ X. B[x, r] (B(x, r), S(x, r)) is a closed ball (an open ball, a sphere)
with center x ∈ (X, ρ) and radius r ≥ 0. M (Int(M ), ∂M ) is the closure (the interior, the boundary) of
a set M ⊂ X. B(X) (B[X], K(X)) is the set of all nonempty bounded (closed, compact) subsets of the
space X. Σ∗N (X) (ΣN (X)) is the set of all (nonempty) subsets in X which consist of (at most) N points.
The elements of ΣN (X) are called N -nets [1]. Let |M W | = inf{|xy| : x ∈ M , y ∈ W } for nonempty
subsets M, W ⊂ X. The map α : B(X) × B(X) → R+ , α(M, W ) = max{β(M, W ), β(W, M )}, where
β(M, W ) = sup{|xW | : x ∈ M }, is the Hausdorff pseudometric on the set B(X) (the restriction of this
pseudometric to B[X] is a metric) ([2], P. 223). Bα (S, r) is an open ball with center S ∈ (Σ∗N (X), α)
and radius r > 0; D(M, W ) = sup{|xy| : x ∈ M , y ∈ W } for M, W ∈ B(X); D(M ) = D(M, M ) is the
diameter of M .
For M ∈ B(X) and a nonempty subset W of X, RW (M ) = inf{β(M, x) : x ∈ W } is the relative Chebyshev radius of M [3]. ZW (M ) = {x ∈ W : β(M, x) = RW (M )} is the set of all relative
Chebyshev centers of M [4]; R(M ) = RX (M ) (Z(M ) = ZX (M )) is the Chebyshev radius (the set
of all Chebyshev centers) of M ; R0 (M ) = RM (M ), Z0 (M ) = ZM (M ); H(M ) = {x ∈ M : β(M, x) =
D(M )} is the set of diametral points of the set M [5]; Q0 (M ) = {x ∈ M : β(Z0 (M ), x) = R0 (M )}.
Let S ∈ ΣN (X), x ∈ S. If S = {x}, then S(x) = S \ {x}. If S = {x}, then S(x) = x. Let
m(S) = min{|xy| : x, y ∈ S, x = y}, m1 (S) = max{|xS(x)| : x ∈ S}, h(S) = {x ∈ S : |xS(x)| =
m(S)},
h1 (S) = {x ∈ S : |xS(x)| = m1 (S)}, d0 (N ) = {S ∈ Σ∗N (X) : D(S) = R0 (S)} be the set of all di∗
= {S ∈ Σ∗N (X) :
ametral N -nets in Σ∗N (X) [5]; Zk,N = {S ∈ Σ∗N (X) : card(Z0 (S)) ≤ k}, Zk,N
card(Z0 (S)) = k} (1 ≤ k ≤ N ), where card(S) is the cardinality of S; md(N ) = {S ∈ Σ∗N (X) :
D(S)=m(S)}, mr0 (N ) = {S∈Σ∗N (X) : m(S) = R0 (S)}, dm1 (N ) = {S∈Σ∗N (X) : D(S) = m1 (S)},
mm1 (N ) = {S ∈ Σ∗N (X) : m(S) = m1 (S)}, m1 r0 (N ) = {S ∈ Σ∗N (X) : m1 (S) = R0 (S)}.
We need the following definitions.
A curve joining points x, y ∈ X whose length equals the distance |xy| between these points is called
a segment [x, y] ([6], P. 42). For λ ∈ [0, 1] and a segment [x, y] ⊂ X, we denote by ωλ [x, y] the point in
*
E-mail: [email protected].
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