  In this paper, the insulated conductivity model with two touching or
close-to-touching inclusions is considered in $\mathbb{R}^{d}$ with $d\geq3$.
We establish the pointwise upper bounds on the gradient of the solution for the
generalized $m$-convex inclusions under these two cases with $m\geq2$, which
show that the singular behavior of the gradient in the thin gap between two
inclusions is described by the first non-zero eigenvalue of an elliptic
operator of divergence form on $\mathbb{S}^{d-2}$. Finally, the sharpness of
the estimates is also proved for two touching axisymmetric insulators,
especially including curvilinear cubes.
