This paper addresses a set of $n$ points $S$ in $\mathbf{R}^d$, where $d \text{ (dimension)} \text{ is greater than or equal to } 2$. We consider a sequence of vertical hyperplanes $H_1,H_2,\text{...},H_{m+1}$ sorted by their first coordinates, ensuring exactly $n/m$ points lie between any two successive hyperplanes. We define $A(S,m)$ as the set of distinct closest pairs in the $\binom{m+1}{2}$ vertical slabs bounded by hyperplanes $H_i$ and $H_j$.
Our main contribution is the construction of a data structure of size $O(n)$, enabling the reporting of the closest pair in the query slab $Q \cap S$ in $O(n^{1/2+\epsilon})$ time. Prior to this work, no linear space data structure with sublinear query time was known.
This research introduces a new perspective on tackling the closest pair problems in high-dimensional spaces, particularly enhancing efficiency in the partitioning and querying of vertical regions.