4 added 55 characters in body

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.

3 added 360 characters in body

This follows from a result of Zalgaller.(If you understand RussianHere is his lecture which inculdes this topic.)

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

2 deleted 5 characters in body; deleted 13 characters in body

This follows from an old a result of Zalgaller. (If you understand Russian, you will enjoy this lectureof Zalgaller.)

1