$\newcommand{\prin}{\mathrm{prin}}$Context: Let $S$ be the unit sphere in some finite dimensional vector space $V$. Given a connected compact Lie group real representation $\rho:G\rightarrow O(V)$, the quotient space $S/G$ carries a smooth stratified Alexandrov space structure. The principal stratum in $S_{\prin}/G$ has a Riemannian structure making it a Riemannian submersion of the principal stratum $S_{\prin}$ in $S$. In [1], the authors show that $S/G$ is not an orbifold if and only if there is a sequence $x_n\in S_{\prin}/G$ converging to $y\in S_{\mathrm{sing}}/G$ a singular orbit such that $\overline\kappa(x_n)\rightarrow \infty$ where $\overline\kappa(x)$ is the maximum sectional curvature at $x$. In other words, the curvature explodes.
I am interested in the stability of metric landmarks in $S/G$. To elaborate, with the induced metric on $S/G$, a metric landmark collection is a set $(z_i)_{i=1}^n$ such that the map $x \mapsto (d(x,z_i))_{i=1}^n$ is injective. In other words, distances to $z_i$ uniquely distinguish points. By a stable metric landmark collection $(z_i)_{i=1}^n$, we mean the existence of $c>0$ such that $$c\cdot d^2(x,y) \leq \sum_{i=1}^n(d(x,z_i) - d(y,z_i))^2$$ for all $x,y\in S/G$. In other words, the map above has a lower Lipschitz bound.
Interest: The question I am interested in is whether every collection of metric landmarks in $S/G$ is stable. If not, when does that take place?
Question: Is it true that curvature explosion implies that some collections of metric landmarks are unstable? Are there any relationships between curvature explosion and stability of landmark collections?
References:
[1] Lytchak, Alexander, and Gudlaugur Thorbergsson. "Curvature explosion in quotients and applications." Journal of Differential Geometry 85.1 (2010): 117-140.
