Ok, I see. Should I try on MO maybe? This question, if I don't solve it, will cost me a good research work. Btw, I see the major problem in $X$ rather than in the dimension of the Euclidean space for some reason. And the main problem, why I doubt the solution exists, is because even for two points something like the above mentioned inequality arises. Moreover, even $X$ were itself a Euclidean space, there needs to be a whole lot of conditions satisfied to build up a piecewise linear function. We would need to separate simplices or something like that ... —
Valery Saharov 9 mins ago