Let $\mathit{Conf}_n(\mathbb{R}^2)$ be the configuration space of $n$ ordered distinct points in the plane. I'd like to know if the topological subspace $C_n$ consisting of points $(p_1,...,p_n)$ with a minimum separation property that every point is of distance at least $1$ from its nearest neighbor is a deformation retract of the original space. Furthermore, is there a deformation retract onto an even smaller, and hopefully compact subspace in which that minimum distance is equal to one? I'd like to say this space is defined so that the graph whose edges correspond to pairs of points satisfying $\lVert p_ip_j\rVert=1$ is connected. This seems like something that ought to appear in robotics, but I could not find a reference or a proof of this statement, if true.

$\begingroup$ $Conf_n$ is a bundle over $Conf_{n1}$ with fiber the $n$punctured plane, so try arguing by induction. You will also need an upper bound on the distances (if you like, you can think of the configurations as living on $S^2$ where the $0$th point is at infinity, so you also want to bound points away from this one). $\endgroup$– Kevin CastoCommented Oct 16, 2023 at 19:54

$\begingroup$ (Basically this comes down to the fact that the $n$punctured plane deformation retracts to a compact set) $\endgroup$– Kevin CastoCommented Oct 16, 2023 at 20:00
1 Answer
Let $\xi : \mathit{Conf}_n(\mathbb R^2)\to \mathbb R$ be function assigning to a configuration the minimum distance between two points. Then $\xi$ is continuous, and we can consider the selfmap of $\mathit{Conf}_n(\mathbb R^2)$ that rescales a configuration $z$ by the factor $\max(1,1/\xi(z))$, resp. $1/\xi(z)$. These provide the deformation retractions you want.
This shows that the distinction between configuration spaces of points and configurations of hard disks only becomes meaningful if the configuration is in a disk or strip of bounded size.