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 $W(x,y; \lambda) \in X$, $\forall x,y \in X$ ?


1 Answer 1


Here is a partial answer to one of the questions. I may expand on the others if I find time (or hopefully someone else provides an answer).

Without additional restrictions, the circle, sphere etc., type positively curved spaces may not be Takahashi convex.

A nice set of spaces that satisfy the condition that you are after would be CAT(0) spaces, for which where $W(x,y,\lambda)$ is the geodesic between $x$ and $y$ parameterized by $\lambda \in [0,1]$.


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