A Takahashi convex metric space is a metric space $(X,d)$ such that there exists a function $W : X \times X \times [0,1] \rightarrow X$ satisfying the condition:
$$d (u, W(x,y; \lambda)) \leq \lambda d(u,x) + (1- \lambda) d(u,y) \quad \forall u \in X \,.$$ 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 type of metric space:
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 $S^{1}$ to be the arc length of $S^1$ between those 2 points.
I wanted to verify if $S^{1}$ is a Takahashi space but I don't know how to find the function $W$!
2. $G : X \rightarrow 2^{X}$ is a KKM map if :
$$\forall A = \lbrace x_{1},...,x_{n} \rbrace \subset X\text{ finite,
}conv \lbrace x_{1},...,x_{n} \rbrace \subset \bigcup\limits_{i=1}^{n} G(x_{i})\,.$$ A convex hull of a set A is defined in a Takahashi convex metric spaces such as :
$$conv(A) = \bigcup\limits_{n \in \mathbb{N}} \tilde{W}^{n}(A)\,, \\\quad\text{ where }\quad\tilde{W}^{n}(A)=\tilde{W}(\tilde{W}^{n-1}(A)), n \geq 2\quad\text{ and }\quad\tilde{W}^{1}(A) = \lbrace W(x,y; \lambda) / x,y \in A, \lambda \in [0,1] \rbrace \,.$$
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 only need the notion of a convex subset, so we only need to verify that $W(x,y; \lambda) \in K$.
Then, what is the role of the Takahashi space definition?
Can't we just define the Takahashi space as $W(x,y; \lambda) \in X$, $\forall x,y \in X$ ?