Definition (triangle):
Let ( M , d ) {\displaystyle (M,d)} be a metric space. A triangle in M {\displaystyle M} is a triple α , β , γ : [ 0 , 1 ] → M {\displaystyle \alpha ,\beta ,\gamma :[0,1]\to M} of continuous functions from the unit interval to M {\displaystyle M} so that α ( 1 ) = β ( 0 ) {\displaystyle \alpha (1)=\beta (0)} , β ( 1 ) = γ ( 0 ) {\displaystyle \beta (1)=\gamma (0)} and > γ ( 1 ) = α ( 0 ) {\displaystyle >\gamma (1)=\alpha (0)} .
Definition ( δ {\displaystyle \delta } -center):
Let δ > 0 {\displaystyle \delta >0} . Let M {\displaystyle M} be a metric space with metric d {\displaystyle d} , and let ( α , β , γ ) {\displaystyle (\alpha ,\beta ,\gamma )} be a triangle in M {\displaystyle M} . A δ {\displaystyle \delta } -center for ( α , β , γ ) {\displaystyle (\alpha ,\beta ,\gamma )} is a point x 0 ∈ M {\displaystyle x_{0}\in M} so that for all t ∈ [ 0 , 1 ] {\displaystyle t\in [0,1]} we have d ( x 0 , α ( t ) ) < δ {\displaystyle d(x_{0},\alpha (t))<\delta } and d ( x 0 , β ( t ) ) < δ {\displaystyle d(x_{0},\beta (t))<\delta } and d ( x 0 , γ ( t ) ) < δ {\displaystyle d(x_{0},\gamma (t))<\delta } .
-center):
. Let 
be a metric space with metric 
, and let 
be a triangle in . A -center for is a point 
so that for all ![{\displaystyle t\in [0,1]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/31a5c18739ff04858eecc8fec2f53912c348e0e5)
we have 
and 
and 
.
![{\displaystyle \alpha ,\beta ,\gamma :[0,1]\to M}](https://wikimedia.org/api/rest_v1/media/math/render/svg/14ee0b5594b2bde5adb9f76a980619dac2fc145e)
