Take the 2-minute tour ×  Encouraged by the progress made in a
recently posted MO problem, here is a "conceptually related"
problem originating from a 2003 joint paper of Sergei Konyagin
and myself.  Suppose we are given $n$ points $z_1,...,z_n$ on
the unit circle $U=\{z\colon |z|=1\}$ and $n$ weights
$p_1,...,p_n\ge 0$ such that $p_1+....+p_n=n$, and we want to
find yet another point $z\in U$ to maximize the product $$
\prod_{i=1}^n |z-z_i|^{p_i}. $$ How large can we make this
product by the optimal choice of $z$?  Conjecture. For any given
$z_1,...,z_n\in U$ and $p_1,...,p_n\ge 0$ with $p_1+...+p_n=n$,
there exists $z\in U$ with $$ \prod_{i=1}^n |z-z_i|^{p_i} \ge 2.
$$  Here are some comments.    • If true, the estimate of the
conjecture is best possible, as evidenced by the situation where
the points are equally spaced on $U$ and all weights are equal
to $1$.    • We were able to resolve a number of particular
cases; say, that where the points $z_i$ are equally spaced on
$U$, and also that where all weight