Distance Geometry

• a set Λ (with at least three distinct elements) is called straight if

for any three elements A, B, and C of Λ holds

$det\; \backslash left(\; \backslash begin\{bmatrix\}$

        0 & d(AB)^2 & d(AC)^2 & 1 \\   d(AB)^2 &    0    & d(BC)^2 & 1 \\   d(AC)^2 & d(BC)^2 &       0 & 1 \\         1 &       1 &       1 & 0  

• a set Π (with at least four distinct elements) is called plane if

for any four elements A, B, C and D of Π holds

$det\; \backslash left(\; \backslash begin\{bmatrix\}$

        0 & d(AB)^2 & d(AC)^2 & d(AD)^2 & 1 \\   d(AB)^2 &    0    & d(BC)^2 & d(BD)^2 & 1 \\   d(AC)^2 & d(BC)^2 &       0 & d(CD)^2 & 1 \\   d(AD)^2 & d(BD)^2 & d(CD)^2 &       0 & 1 \\         1 &       1 &       1 & 1       & 0  

but not all triples of elements of Π are straight to each other;

• a set Φ (with at least five distinct elements) is called flat if

for any five elements A, B, C, D and E of Φ holds

$det\; \backslash left(\; \backslash begin\{bmatrix\}$

        0 & d(AB)^2 & d(AC)^2 & d(AD)^2 & d(AE)^2 & 1 \\   d(AB)^2 &    0    & d(BC)^2 & d(BD)^2 & d(BE)^2 & 1 \\   d(AC)^2 & d(BC)^2 &       0 & d(CD)^2 & d(CE)^2 & 1 \\   d(AD)^2 & d(BD)^2 & d(CD)^2 &       0 & d(DE)^2 & 1 \\   d(AE)^2 & d(BE)^2 & d(CE)^2 & d(DE)^2 &       0 & 1 \\         1 &       1 &       1 & 1       &       1 & 0  

but not all quadruples of elements of Φ are plane to each other;

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