Firstly I am not sure if the definition of "the triangle principle" is used correctly.
What I actually want to express is, in a 3D space, if point $\mathbf{X}$ is an interior point of some other points $\mathbf{X}_1, \mathbf{X}_2,\cdots,\mathbf{X}_m$ (points $\mathbf{X}_i, i=1,2,\cdots,m$ are vertices of a convex hull which contains $\mathbf{X}$), then is there any distance measurement metric under which the relationship $$\mathrm{dist}(\mathbf{X}_0, \mathbf{X})\le\operatorname*{\mathrm{max}}_{i=1}^{m}\{\mathrm{dist}(\mathbf{X}_0, \mathbf{X}_i)\}$$ is not satisfied?
Apparently under some homogenous distance metric, e.g. Euclidean distance, this relationship will always be true, but what if under other non-homogenous metric? Like CIEDE2000 color difference? in which the distance between two points in XYZ color space is position-dependent, since the XYZ space is not uniform for human's color perception.
So, under a non-homogenous distance measurement metric, how to prove the above relationship is satisfied or not?
Many thanks to your all.