# Maximize the intersection of a n-dimensional sphere and an ellipsoid.

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?

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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.