What type of set is the intersection of two spheres? 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The given link is no longer active but I have found the article: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
