Let $K$ be a convex body of volume 1 in $\mathbb{R}^n$ and $x$ a (variable) point on the boundary of $K$. Define $f_K(x)$ to be the volume of the convex hull of the union of $K$ with its reflection in $x$, that is,

$$f_K(x)=\rm{Vol}\left\{\rm{Conv}\left[K\cup(2x-K)\right]\right\}.$$

It is easy to see that for an $n$-dimensional ellipsoid $E$, the function $f_E(x)$ is constant, $f_E=e_n$, depending on $n$ only:

$\displaystyle e_n=1+{{2\sigma_{n-1}\over{\sigma_n}}}$, where $\sigma_k$ denotes the volume of the $k$-dimensional unit ball.

Question 1. If $f_K(x)$ is a constant function, must $K$ be an ellipsoid?

Question 2. Is $\displaystyle e_n=\max_K\left[\,\min_{x\,\in\, bdK}\,f_K(x)\right]$? (Here the maximum is taken over all $n$-dimensional convex bodies $K$ of volume 1.)

If the answer to either of these questions is known, a reference would be greatly appreciated.