  This paper is concerned with an explicit value of the embedding constant from
$W^{1,q}(\Omega)$ to $L^{p}(\Omega)$ for a bounded domain
$\Omega\subset\mathbb{R}^N~(N\in\mathbb{N})$, where $1\leq q\leq p\leq \infty$.
To obtain this value, we previously proposed a formula for estimating the
embedding constant on bounded and unbounded Lipschitz domains by estimating the
norm of Stein's extension operator, in the article (K. Tanaka, K. Sekine, M.
Mizuguchi, and S. Oishi, Estimation of Sobolev-type embedding constant on
domains with minimally smooth boundary using extension operator, Journal of
Inequalities and Applications, Vol. 389, pp. 1-23, 2015). This formula is also
applicable to a domain that can be divided into Lipschitz domains. However, the
values computed by the previous formula are very large. In this paper, we
propose several sharper estimations of the embedding constant on a bounded
domain that can be divided into convex domains.
