Consider the metric space (M,d) where set M contains sufficiently many (at least five) distinct elements,
and consider the assignment cf of coordinates to (the elements of) set M,
cf:M↔R3;cf[P]:={xP,yP,zP}
such that distinct coordinates values are assigned to distinct elements of set M, and
such that for the function
f:(R3×R3)→R;
f[{xP,yP,zP},{xQ,yQ,zQ}]:= √(xQ−xP)2+(yQ−yP)2+(zQ−zP)2≡√∑k∈{xyz}(kQ−kP)2
and for any three distinct elements A, B, and J ∈M holds
f[cf[A],cf[J]]d[B,J]=f[cf[B],cf[J]]d[A,J].
Is the metric space (M,d) therefore flat?
(i.e. in the sense of vanishing Cayley-Menger determinants of distance ratios between any five elements of set M.)
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