Convex metric

In mathematics, convex metric spaces are, intuitively, metric spaces with the property any "segment" joining two points in that space has other points in it besides the endpoints.

Formally, consider a metric space (X, d) and let x and y be two points in X. A point z in X is said to be between x and y if all three points are distinct, and

that is, the triangle inequality becomes an equality. A convex metric space is a metric space (X, d) such that, for any two distinct points x and y in X, there exists a third point z in X lying between x and y.

Metric convexity:

Let ${\displaystyle (X,d)}$ be a metric space (which is not necessarily convex). A subset ${\displaystyle S}$ of ${\displaystyle X}$ is called a metric segment between two distinct points ${\displaystyle x}$ and ${\displaystyle y}$ in ${\displaystyle X,}$ if there exists a closed interval ${\displaystyle [a,b]}$ on the real line and an isometry

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