  Let $G$ be a group and $H_1$,\ldots,$H_s$ be subgroups of $G$ of indices
$d_1,\ldots,d_s$ respectively. In 1974, M. Herzog and J. Sch\"onheim
conjectured that if $\{H_i\alpha_i\}_{i=1}^{i=s}$, $\alpha_i\in G$, is a coset
partition of $G$, then $d_1,\ldots,d_s$ cannot be distinct. In this paper, we
present the conjecture as a problem on vanishing sum of roots of unity and
convex polygons and prove some results using this approach.
