On decomposition of neighborhood of a circular cone related to principal curvatures

We give an alternative proof of a result on the uniform overlap of the algebraic sums of the sets arising from a decomposition of a neighborhood of a circular cone in ${\mathbb{R}}^3$. It is known that the uniform overlap result can be applied to make a unified approach for the proofs of a theorem on the maximal Bochner-Riesz operator on ${\mathbb{R}}^2$ and a theorem on the maximal spherical means on ${\mathbb{R}}^2$.