**(1)** $\mathbb{R}^2$.
I'd like to place $n$ points in the plane so that the smallest angle they
determine is as large as possible.
In a sense, such a point set is in very general position, not only
avoiding three points collinear, but also avoiding near collinearities.
Define the smallest angle of a set $S$ of points to the be smallest angle
of any triangle formed by three points in $S$. So the $n=4$ and $n=5$ point
sets shown below have smallest angles $45^\circ$ and $36^\circ$ respectively.
<hr />
[![PtsNoSmallAngs][1]][1]
<hr />
> ***Q1***. What is the maximum of the smallest angle determined by any set $S$ of $n$ points, the maximum over all $S$? Is $S$ the vertices of a regular $n$-gon?
*Update*. Answered *Yes* by fedja with a nice proof in the comments.
**(2)** $\mathbb{R}^3$ (*Added*).
In 3D, the optimal arrangement seems to be akin to
packing points on a sphere, e.g., the
[Tammes problem or the Thompson problem](https://mathoverflow.net/a/212556/6094).
Below shows the smallest angle
realized by the $12$ vertices of the icosahedron.
<hr />
[![IcosaSmallAng][2]][2]
<br />
<sup>
Smallest angle $\approx 31.7^\circ$.
</sup>
<hr />
> ***Q2***. The same question in $\mathbb{R}^3$, and in $\mathbb{R}^d$, $d>3$.
Likely this question has been
studied, in which case pointers to the literature would be appreciated.
[1]: https://i.sstatic.net/bM6cQ.jpg
[2]: https://i.sstatic.net/A70rs.jpg