Monotone Simultaneous Embeddings of Paths in R^d

TL;DR

The paper investigates the possibility of embedding multiple paths in Euclidean space such that each path is monotone, proving that in any dimension d ≥ 2, certain sets of d+1 paths cannot be simultaneously embedded with monotonicity.

Contribution

It establishes a fundamental limitation on monotone simultaneous embeddings of paths in Euclidean spaces of dimension two or higher.

Findings

For any dimension d ≥ 2, there exists a set of d+1 paths that cannot be embedded monotonically.

The result generalizes previous understanding of geometric embeddings and monotonicity constraints.

Provides a theoretical boundary for embedding multiple paths with monotonicity in higher dimensions.

Abstract

We study the following problem: Given $k$ paths that share the same vertex set, is there a simultaneous geometric embedding of these paths such that each individual drawing is monotone in some direction? We prove that for any dimension $d \geq 2$, there is a set of $d+1$ paths that does not admit a monotone simultaneous geometric embedding.