  In this paper, we propose new lower and upper bounds on the linear extension
complexity of regular $n$-gons. Our bounds are based on the equivalence between
the computation of (i) an extended formulation of size $r$ of a polytope $P$,
and (ii) a rank-$r$ nonnegative factorization of a slack matrix of the polytope
$P$. The lower bound is based on an improved bound for the rectangle covering
number (also known as the boolean rank) of the slack matrix of the $n$-gons.
The upper bound is a slight improvement of the result of Fiorini, Rothvoss and
Tiwary [Extended Formulations for Polygons, Discrete Comput. Geom. 48(3), pp.
658-668, 2012]. The difference with their result is twofold: (i) our proof uses
a purely algebraic argument while Fiorini et al. used a geometric argument, and
(ii) we improve the base case allowing us to reduce their upper bound $2
\left\lceil \log_2(n) \right\rceil$ by one when $2^{k-1} < n \leq
2^{k-1}+2^{k-2}$ for some integer $k$. We conjecture that this new upper bound
is tight, which is suggested by numerical experiments for small $n$. Moreover,
this improved upper bound allows us to close the gap with the best known lower
bound for certain regular $n$-gons (namely, $9 \leq n \leq 13$ and $21 \leq n
\leq 24$) hence allowing for the first time to determine their extension
complexity.
