SCHWEITZER 1967 Problem 10

Let $\sigma(S_n,k)$ denote the sum of the $k$ th powers of the lengths of the sides of the convex $n$ -gon $S_n$ inscribed in a unit circle. Show that for any natural number greater than $2$ there exists a real number $k_0$ between $1$ and $2$ such that $\sigma(S_n,k_0)$ attains its maximum for the regular $n$ -gon.

L. Fejes Toth