Talk:Equilateral dimension

Unclear statement

The section Riemannian manifolds reads as follows:

"For any $d$-dimensional Riemannian manifold the equilateral dimension is at least $d+1$. For a $d$-dimensional sphere, the equilateral dimension is $d+2$, the same as for a Euclidean space of one higher dimension into which the sphere can be embedded. At the same time as he posed Kusner's conjecture, Kusner asked whether there exist Riemannian metrics with bounded dimension as a manifold but arbitrarily high equilateral dimension."

It is entirely unclear from this vague wording what the last sentence refers to.

What is the "dimension" of a Riemannian metric?

Is it asking about a fixed smooth manifold with various Riemannian metrics on it?

Or is it asking about various smooth manifolds with various Riemannian metrics on them? 2601:200:C000:1A0:986D:4E1A:4FA6:7EE (talk) 17:32, 21 February 2022 (UTC)

Guy's questions answered

Re the equilateral dimension of Riemannian manifolds, see my preprint https://arxiv.org/abs/2401.06328 and blog post https://11011110.github.io/blog/2024/01/22/equilateral-dimension-riemannian.html (not yet even submitted anywhere let alone reliably published). In short, Riemannian 2-spheres have bounded equilateral dimension; incomplete Riemannian disks, complete Riemannian 2-manifolds of infinite genus, and Riemannian metrics on $\mathbb {R} ^{3}$ do not. —David Eppstein (talk) 02:25, 23 January 2024 (UTC)