It is shown here that for every n ∈ ℕ, any embedding into L1 of the n-fold Pythagorean power of the n-dimensional Hamming cube incurs distortion that is at least a constant multiple of √n. This is achieved through the introduction of a new bi-Lipschitz invariant of metric spaces that is inspired by a linear inequality of Kwapień and Schütt (1989). The new metric invariant is evaluated here for L1, implying the above nonembeddability statement. Links to the Ribe program are discussed, as well as related open problems.
All Science Journal Classification (ASJC) codes
- Algebra and Number Theory
- Geometry and Topology
- Metric embeddings
- Ribe program