I have the conjecture that the volume of the intersection between an $n$-dim sphere (of radius $r$) and an ellipsoid (with one semi-axis larger than $r$) is maximized when the two are concentric, but still did not find a way to prove it. Any suggestion? 



Maximize the intersection of a n-dimensional sphere and an ellipsoid.
From a [result of Zalgaller](http://mi.mathnet.ru/znsl686), this is true for any two centrally symmetric bodies.
([Here](http://www.mathnet.ru/php/seminars.phtml?option_lang=rus&presentid=169) is his lecture which inculdes this topic.)


Namely, assume that the center of first body is at $0$.
If $\vec r$ is the center of the secoind body and $v(\vec r)$ is the volume of intersection with the first one
then according to Zalgaller's theorem $v>0$ in a convex domain and inside of this domain the function $v^{1/n}$ is concave.
Clearly $\vec r=0$ is an extremal point of $v$.
Thereofre $0$ is the point of maximum.


