A Generalization of Blaschke's Convergence Theorem in Metric Spaces

A metric space

together with a set-valued mapping

G: X\times X\to 2^X G : X × X → 2 X

is said to be a generalized segment space

(X, d, G) ( X , d , G )

if

G(x, y)\not=\emptyset G ( x , y ) ≠ ∅

for all

x, y\in X x , y ∈ X

and for any sequences

x_n\to x x n → x

and

y_n\to y y n → y

in

X X

,

d_H\big(G(x_n, y_n), G(x, y) \big) \to 0 d H ( G ( x n , y n ) , G ( x , y ) ) → 0

as

n\to \infty n → ∞

, where

d_H d H

is the Hausdorff distance. Normed linear spaces, nonempty convex sets, and proper uniquely geodesic spaces, etc are generalized segment spaces for suitable

G G

. A subset

A A

of

X X

is called

G G

-type convex if

G(x, y)\subset A G ( x , y ) ⊂ A

whenever

x, y\in A x , y ∈ A

. We prove a generalization of Blaschke's convergence theorem for metric spaces: if

(X, d, G) ( X , d , G )

is a proper generalized segment space, then every uniformly bounded sequence of nonempty

G G

-type convex subsets of

X X

contains a subsequence which converges to some nonempty compact

G G

-type convex subset in

X X

.