Consider a Lie group $G$ acting properly on a manifold $M$. Then by the slice theorem we can find for any point $m\in M$ a submanifold transverse to the orbit $\mathcal{O}$ through $m$ and which is (essentially) an open ball around the origin of the fibre $N_m\mathcal{O}$ of the normal bundle to $\mathcal{O}$, on which the stabiliser of $m$ acts linearly (I'm clearly compressing and glossing over a lot here - a reference is this).
I'd like to say something about a sort of 'intersection' of slices, and not the literal intersection. Take a collection of slices $S_i$ such that the saturations $sat(S_i)$ are an open cover of $M$. This is preordered by dimension of the slices, equivalently by codimension of the orbits about which they are centred. Given two slices $S_i,S_j$ with $i \leq j$, let $S_{ij} :=S_i \cap sat(S_j)$. This is the intersection of a tubular neighbourhood with a slice of a tubular neighbourhood, and I'd like to know if it is convex, when considered as a subset of $N_m\mathcal{O}$. And I would like this true for any $S_{ij}$, and possibly for more general $S_{ijk\ldots} = S_i \cap sat(S_j)\cap sat(S_k) \cap \ldots$ for $i\leq j\leq k \ldots$.
Now this might not always be true for a given slices, but it may be true after taking another slice $S'_i$ along the same orbit, or one might have to take smaller slices (and correspondingly more of them to ensure every orbit is covered).
Are there any results in the literature along these lines? I can imagine one might proceed by taking $G$-invariant metrics on the normal bundles $N\mathcal{O}_i$ and taking some estimates on something like a transverse convexity radius, but I don't have any serious computation to back this up.
A cute example when it is true: take the trivial group action $\langle e\rangle \times M \to M$, then a collection of slices with the properties as above is something like a good open cover.
