real-projective-space-proves-2m-bound-sharp

IN premise — summaries/2026/08/24/wiki-Whitney_embedding_theorem.md

Created 2026-08-24T17:11:28+00:00

Real projective m-spaces (for m a power of 2) do not embed in R^(2m-1), proving that the Whitney strong embedding bound of R^(2m) cannot be improved in general.

Summary

There are geometric shapes (real projective spaces built from powers of two) that are too "twisted" to be represented in any space one dimension smaller than Whitney's bound allows. This means Whitney's result is not just a rough estimate but a genuine ceiling: no general method can squeeze these objects into fewer dimensions, so any system relying on embedding dimensionality must plan for the full 2m dimensions in the worst case.