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We give an alternative proof of a result on the uniform overlap of thealgebraic sums of the sets arising from a decomposition of a neighborhood of acircular cone in $\Bbb R^3$. It is known that the uniform overlap result can beapplied to make a unified approach for the proofs of a theorem on the maximalBochner-Riesz operator on $\Bbb R^2$ and a theorem on the maximal sphericalmeans on $\Bbb R^2$.

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