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.

Q. 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?

The same question may be asked in $\mathbb{R}^d$, $d>2$. Likely this question has been studied, in which case pointers to the literature would be appreciated.

(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.

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. Below shows the smallest angle realized by the $12$ vertices of the icosahedron.

Likely this question has been studied, in which case pointers to the literature would be appreciated.