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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 1a. If $f_K(x)$ is a constant function, must $K$ be centrally symmetric?

Question 1b. If $K$ is centrally symmetric and $f_K(x)$ 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.

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    $\begingroup$ The case $n=2$ may be more tractable -- what happens there? $\endgroup$ – Noam D. Elkies Nov 19 '17 at 22:00
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    $\begingroup$ I believe the answer is "yes"; the lower the dimension the stronger my belief. $\endgroup$ – Wlodek Kuperberg Nov 19 '17 at 22:03
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    $\begingroup$ May I ask you to expand a bit on "It is easy to see..."? $\endgroup$ – Joseph O'Rourke Nov 20 '17 at 0:00
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    $\begingroup$ The question is invariant under linear transformations, so it's enough to do the case of a sphere, which is clear by symmetry. $\endgroup$ – Noam D. Elkies Nov 20 '17 at 0:20
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Not an answer, just an illustration.


          ThreeEllipses
          Three $2 \times 1$ ellipses, rotated $30^\circ$, reflected through different points. Hulls have the same area.


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  • $\begingroup$ Nice drawing. This works for ellipsoids in every dimension. $\endgroup$ – Wlodek Kuperberg Nov 20 '17 at 1:16
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    $\begingroup$ These are nice drawings — but I’d like them better as edits to the post than as an answer. $\endgroup$ – Matt F. Nov 20 '17 at 19:03
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There are some partial answers in this paper

A.G. Horváth, Z.Lángi, On the volume of the convex hull of two convex bodies, Monat. fur Mathematic 174, pp 219-229 https://link.springer.com/article/10.1007/s00605-013-0526-x

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