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Let $Y$ denote the one-point union of three unit intervals. Since $Y$ embeds isometrically in the plane, it follows that the nth cartesian power $Y^n$ embeds isometrically — and also topologically — in $\mathbb R^{2n}$.

The metric on $Y^n$ is the cartesian power of the intrinsic metric on $Y$, so that any subset that is a product of n unit intervals has the natural metric of the unit n-cube $[0, 1]^n$.

By an "isometric embedding" of $Y^n$ into the euclidean space $\mathbb R^{k}$ I mean that $Y^n$ is isometric to some subset $X$ ⊂ $\mathbb R^{k}$ where $X$ is endowed with its intrinsic metric: D(p,q) = the infimum of the lengths of all paths from p to q within the space $X$.

If $t(n)$ is defined as the lowest dimension that $Y^n$ embeds in topologically, and $m(n)$ as the lowest dimension that $Y^n$ embeds in isometrically, then clearly we have $t(n) \le m(n) \le 2n$.

What else is known about t(n) and m(n)?

Dan Asimov
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    Might want to clarify what you mean by “isometrically” here. Since $Y^n$ is not a manifold, are we just putting an intrinsic metric on $Y$, and asking for an embedding that preserves path length, rather than distances? (The embedding of a tripod with its intrinsic metric into the plane is not an isometry in the sense of preserving distances) Also what metric are you putting on the product? – M W Feb 02 '24 at 14:59
  • Good point. I will clarify this ambiguity. – Dan Asimov Feb 02 '24 at 15:01
  • From what I remember, $t(Y^n)=2n$. I will try to find a reference (I think, it was proven by van Kampen). – Moishe Kohan Feb 02 '24 at 15:21

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One can show that $t(Y^n)=2n$. For instance, you can use the fact that the $n$-fold join of the 3-point set does not embed in $R^{2n-2}$ due to the van Kampen obstruction, combined with Lemma 8 in

Bestvina, Mladen; Kapovich, Michael; Kleiner, Bruce, Van Kampen’s embedding obstruction for discrete groups, Invent. Math. 150, No. 2, 219-235 (2002). ZBL1041.57016.

I am quite sure it was known before and most likely proven in Wu's book "A theory of imbedding, immersion, and isotopy of polytopes in a euclidean space." (I do not have that book.) It might be also in van Kampen's original 1933 paper, but I do not have an access to it either and van Kampen's proofs were, unsurprisingly, faulty.

Moishe Kohan
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  • Thanks, Moishe! I also found a 2018 version of this paper in the arXiv: https://arxiv.org/abs/math/0010141. – Dan Asimov Feb 02 '24 at 16:58