This follows from
From a result of Zalgaller, this is true for any two centrally symmetric bodies. (Here is his lecture which inculdes this topic.)
Namely, assume that the center of ellipse first body is at $0$. If $\vec r$ is the center of the ball secoind body and $v(\vec r)$ is the volume of intersection with the ellipse 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.
