Is there a curve of constant width $1$ on which it is impossible to arrange the five points $A, B, C, D, E$ so that $\max(AB, BC, CD, DE, EA) \leqslant \sin (\frac{\pi}{5})$?
For example, on a circle, you can place the points in the form of a regular pentagon, and the distance will be exactly $\sin (\frac{\pi}{5})$.
Does anyone have any idea how to place such points on an arbitrary curve of width $1$?
And more general hypothesis: for any $n$ on any curve of constant width $1$, we can place $n$ points $A_1, A_2, ..., A_n, A_{n+1}=A_1$, such that $\max_{i=1..n} (A_{i}A_{i+1}) \leqslant \sin (\frac{\pi}{n})$.
Upd: this is true for $n \leqslant 4$. $n=1$ and $n=2$ is trivial cases. For $n = 3$, we can put the curve in a universal Pal's cover and then divide it into three parts of diameter $\sin(\frac{\pi}{3})$. For $n=4$, we can put the curve in a square with side $1$, and divide it into four small squares of diameter $\sin(\frac{\pi}{4})$.