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:

<a href="https://i.sstatic.net/P0L0B.jpg"><img src="https://i.sstatic.net/rfunW.png" title="source: imgur.com" /></a>

$\displaystyle e_n=1+{{2\sigma_{n-1}\over{\sigma_n}}}$,  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.