A Takahashi convex metric space is a metric space $(X,d)$ such that $\exists W : X \times X \times [0,1] \rightarrow X$ that satisfies :

$d (u, W(x,y; \lambda)) \leq \lambda d(u,x) + (1- \lambda) d(u,y)$; $\forall u \in X$.

A subset $K \subset X$ is convex if $W(x,y; \lambda) \in K$, $\forall x,y \in K$ and $\lambda \in [0,1]$.

I have some questions about this metric type spaces:

(1) I wanted some examples of Takahashi metric spaces which are not linear . I've started thinking about the unit circle $S^{1} = \lbrace (x,y) \in \mathbb{R}^{2} / x^{2}+y^{2}=1 \rbrace$.

I defined the metric between 2 points in $S^{1}$ such that it's arc length between those 2 points.

I wanted to verify if $S^{1}$ is a Takahashi space but i don't know how can i find the function W !

(2) $G : X \rightarrow 2^{X}$ is a KKM map if :

$\forall A = \lbrace x_{1},...,x_{n} \rbrace \subset X$ finite,
$conv \lbrace x_{1},...,x_{n} \rbrace \subset \bigcup\limits_{i=1}^{n} G(x_{i})$.

A convex hull of a set A is defined in a Takahashi convex metric spaces such as :

$conv(A) = \bigcup\limits_{n \in \mathbb{N}} \tilde{W}^{n}(A)$, where $\tilde{W}^{n
}(A)=\tilde{W}(\tilde{W}^{n-1}(A)), n \geq 2$ and $\tilde{W}^{1}(A) = \lbrace W(x,y; \lambda) / x,y \in A, \lambda \in [0,1] \rbrace$.

How can i find the convex hull in a particular case of a non linear space, if i can't define the function W ?

(3) For the convex hull of K, we need only the notion of convex subset so we only need $W(x,y; \lambda) \in K$.

Then, what is the role of the Takahashi space definition?

Can't we just define the Takahashi space as $W(x,y; \lambda) \in X$, $\forall x,y \in X$ ?