An example of Vietoris–Rips complex

Question

The below figure is from https://arxiv.org/abs/1710.04019. I'm new to the concept of the simplicial complex and studying it through the article. The given definition of Vietoris–Rips complex $\text{Rips}_{\delta}(X)$ of the points in $X$ is "the set of simplicies $[x_0, \cdots, x_k]$ such that $d(x_i, x_j) \le \delta$ for all $(i, j)$" where $d$ is a metric. I cannot understand why the Vietoris–Rips complex with radius $2 \alpha$ in this figure (the right one) has the tetrahedron. The distance between the upper point and the bottom point of the tetrahedron is larger than $2 \alpha$, and so they cannot be in a simplex. I think the bottom part should be the same with the left one. Do I misunderstand something?

It is just an inaccurate picture, whose main goal was to illustrate the difference between Chech and Rips. — Moishe Kohan, Dec 22 '17 at 10:38

score 3 · Accepted Answer · answered Dec 23 '17 at 02:33

3

You are right and the authors of the paper are just sloppy with their pictures.

answered Dec 23 '17 at 02:33

Moishe Kohan

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@madprob: What I wrote, does provide a full answer. – Moishe Kohan Dec 23 '17 at 03:06
Sorry! You are right. – madprob Dec 23 '17 at 04:15

An example of Vietoris–Rips complex

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