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Suppose $U$ is an open subset of $\mathbb{R}^n$ which is star shaped with respect to $p\in U$. I'll call $p$ a midpoint of $U$ if for any line $\ell$ through $p$, the point $p$ is the midpoint of the line segment $\ell \cap \overline{U}$. I allow the case that the bi-infinite line through $p$ is contained in $U$; forgive me for abusing the term midpoint in that regard.

First off, does this go by some other name? I'd be stunned if this wasn't a classical topic in geometry, but a cursory google search didn't turn up anything.

In any case, it's pretty obvious that there are many examples of this type of set, although other than interiors of spheres and ellipsoids and all of $\mathbb{R}^n$, I don't really know if they have a preferred name. The examples I've imagined look "especially like stars" if that means anything. In the case of $U\subset\mathbb{R}^2$ with $C^2$-boundary, a believable (I have not proved this yet) necessary and sufficient condition is the following. Suppose $x,y\in \partial{U}$ lie on a line segment through $p$. Then the geodesic curvature $\kappa$ satisfies $\kappa(x)=\kappa(y)$. Perhaps the correct thing in higher dimensions is the agreement of all of the principal curvatures? One other basic observation is that the distance function from $p$ to the boundary of $U$ denoted $d:\partial U \rightarrow \mathbb{R}$ descends to a function on the projective space $\mathbb{RP}^{n-1}$.

My question: does anyone know of some literature on this subject?

Thanks for any help.

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    $\begingroup$ Is there a difference between having a midpoint, and being symmetric about a point? That might answer your preliminary question. $\endgroup$ Aug 15, 2014 at 16:44
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    $\begingroup$ Origin symmetric star-shaped sets have been called centred in this Molchanov paper: arXiv abs link $\endgroup$ Aug 15, 2014 at 17:04

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