  We study the insulated conductivity problem with inclusions embedded in a
bounded domain in $\mathbb R^n$, for $n \ge 3$. The gradient of solutions may
blow up as $\varepsilon$, the distance between inclusions, approaches to $0$.
We established in a recent paper optimal gradient estimates for a class of
inclusions including balls. In this paper, we prove such gradient estimates for
general strictly convex inclusions. Unlike the perfect conductivity problem,
the estimates depend on the principal curvatures of the inclusions, and we show
that these estimates are characterized by the first non-zero eigenvalue of a
divergence form elliptic operator on $\mathbb S^{n-2}$.
