Given 4 points $A$, $B$, $C$ and $D$ in general position in the euclidean plane, is it possible to determine from the 6 distances $AB$, $BC$, $CD$, $AD$, $AC$ and, $BD$ alone, whether every point is a corner of their convex hull under the restriction that comparing sums and/or differences of the distances is the only allowed operation and that no information about the point's coordinates is available. 


Background of the question is whether it is possible to generalize the concept of intersection to non-adjacent edges of a general weighted graph.
If the answer were affirmative, then geometric concepts like inside/outside relations or convex hulls would be an "intrinsic" property of graphs; triangulations of complete graphs would then also be more flexible and could contain complete subgraphs of order 4.



Characterizing Convex Configurations of Quadrupels of Coplanar Points via Linear (In-)equalities between Distance Sums or Differences
Here is a sketch of an answer that involves the *squared* Euclidean distances between the four points $A,B,C,D$ (and only the squared ones, not a mixture between squared and non-squared distances.)


Suppose $D$ is the origin, and consider the vectors $a=DA, b=DB, c=DC$. From
$$AB^2 = \langle a-b,a-b \rangle = DA^2 -2\langle a,b\rangle + DB^2,$$ we get $\langle a,b\rangle$ as a linear expression in terms of the squared distances, and similarly for the other mixed inner products. The inner "self"-products like $\langle a,a\rangle=DA^2$ are directly available anyway.
Assuming that $a$ and $b$ are linearly independent, we can write $$c=\lambda a + \mu b.\ \ \ \ \ (1)$$ Then $D$ (the origin) is in the convex hull of $A,B,C$ iff $\lambda\le 0$ and $\mu \le 0$. Multiplying (1) by $a$, $b$, and $c$ gives an overdetermined system of three equations
$$\langle a,c\rangle = \lambda\langle a,a\rangle+\mu\langle a,b\rangle,$$ 
etc. from which an expression (actually, three equivalent expressions) for $\lambda$ and $\mu$ can be derived using Cramer's rule, as a quotient of determinants. Hence, the sign of these determinants tells us if $D$ is a vertex of the convex hull or not.


So in the end it boils down to $2\times 2$ determinants whose entries are linear in the squared distances, i.e., degree-2 polynomials in the squared distances. One would have to work out what these expressions are. Who knows, maybe a clever combination of the various redundant expression gives even linear expressions, or expressions that can be factorized into nice linear terms.


