Let $X$ be a normed space and $S(x, \varepsilon)=\{y\in X: \|x-y\|=\varepsilon\}$ denote the sphere centered at $x\in X$ with radius $\varepsilon>0.$ Consider two different spheres $S_1=S(x_1, \varepsilon_1)$ and $S_2=S(x_2, \varepsilon_2)$ with non-empty intersection. What can we say about the set $S_1\cap S_2$? Is it true that this set is always homeomorphic to a sphere or a ball of "one dimension less" (or even more dimensions less)?
What we know for sure: A paper by Jussi Vaisala contains a short proof of the following:
Lemma 2.2: The intersection of two spheres in a finite dimensional normed space $X$ of dimension $n\geq 3$ is a connected set. Furthermore, if $X$ is strictly convex, then this intersection is either a singleton or homeomorphic to an $(n-2)$-sphere.
My guess is that the second claim of the Lemma still holds for $n=2$, but was omitted from it because of the intersection not being connected. So, this should pretty much settle my question for finite dimensional strictly convex spaces.
Without the assumption of $X$ being strictly convex we may have some more possibilities. Consider $X=\mathbb{R}^3$ equipped with the $\|⋅\|_\infty$ norm. The spheres are in fact cubes in $\mathbb{R}^3$ and the intersection of two cubes may additionally be (homeomorphic to) a square, which is the closed unit ball of $(\mathbb{R}^2, \|⋅\|_\infty)$, or (homeomorphic to) a line segment, which is the closed unit ball of $(\mathbb{R}, \|⋅\|_\infty)$.
So, what can we say if we remove the strict convexity assumption? Is there a counterexample where the intersection is neither a sphere or a ball? I couldn't find any. Additionally what can be said for infinite dimensional normed spaces?
I couldn't find any literature regarding this topic and probably for a good reason since Vaisala mentions that while $S_1\cap S_2$ being connected was known by Novikoff in 1955, $S_1\cap S_2$ being homeomorphic to the $(n-2)$-sphere seemed to be a new result [sic] at the time he published it (2010).
The given link is no longer active but I have found the article:
MR2971762 Väisälä, Jussi Regular simplices in three-dimensional normed spaces. Beitr. Algebra Geom. 53 (2012), no. 2, 569–570.
https://doi.org/10.1007/s13366-012-0098-2