On prescribing total orders for bipartite sets of distances in the Euclidean Plane

Abstract

In this note we give a negative answer to a question proposed by Almendra-Hernández and Martínez-Sandoval. Let n ≤ m be positive integers and let X and Y be sets of sizes n and m in Rn-1 such that XUY is in generic position. There is a natural order on XxY induced by the distances between the corresponding points. The question is if all possible orders on XxY can be obtained in this way. We show that the answer is negative when n<m. The case n=m remains open.