General embedding problems and two-distance sets in Minkowski planes (Q945968)

A metric space \((D,\rho)\) is a \(2\)-distance set if \(\{\rho(x,y): x,y \in D\), \(x\neq y\}\) has at most two elements. The question arises of which \(2\)-distance sets may be isometrically embedded in the Euclidean plane (there are six maximal ones, five with 4 points and the regular pentagon) or some other normed plane. For this \(D\) must be finite and may be taken to be \([n]:=\{1,2,\dots,n\}\). The author studies the general problems of whether or not there is an embedding of a given metric space \(([n],\rho)\) into a suitable Minkowski space of dimension \(d \geq 1\) and, if so, to describe all such embeddings. ``The solution of the problem has both an algebraic part (analytical geometry) as well as a discrete part. The reason for this is that for each of finitely many combinatorial candidates, characterized by the relative position of the points and the distinction between large and small distances, the problem can be transformed into a system of polynomial equations and inequalities whose unknown variables are geometric coordinates and the occurring distances. Both parts together were handled with the use of a computer program, using some evolved external libraries and systems''. In the case \(d=2\) the author is able to use the computer program to give a complete solution. There are eleven maximal \(2\)-distance sets that may be embedded in some Minkowski plane; one with 4 points, four with 5, four with 6, one with 7 and one with 9. The last is the set \(\{(0,0), (\pm 1, 0), (0, \pm 1), (\pm1,\pm 1)\}\) in \(\ell^2_{\infty}\) whose distances are \(1\) and \(2\).