  Range closest-pair (RCP) search is a range-search variant of the classical
closest-pair problem, which aims to store a given set $S$ of points into some
space-efficient data structure such that when a query range $Q$ is specified,
the closest pair in $S \cap Q$ can be reported quickly. RCP search has received
attention over years, but the primary focus was only on $\mathbb{R}^2$. In this
paper, we study RCP search in higher dimensions. We give the first nontrivial
RCP data structures for orthogonal, simplex, halfspace, and ball queries in
$\mathbb{R}^d$ for any constant $d$. Furthermore, we prove a conditional lower
bound for orthogonal RCP search for $d \geq 3$.
