  In this paper we present new data structures for two extensively studied
variants of the orthogonal range searching problem.
  First, we describe a data structure that supports two-dimensional orthogonal
range minima queries in $O(n)$ space and $O(\log^{\varepsilon} n)$ time, where
$n$ is the number of points in the data structure and $\varepsilon$ is an
arbitrarily small positive constant. Previously known linear-space solutions
for this problem require $O(\log^{1+\varepsilon} n)$ (Chazelle, 1988) or
$O(\log n\log \log n)$ time (Farzan et al., 2012). A modification of our data
structure uses space $O(n\log \log n)$ and supports range minima queries in
time $O(\log \log n)$. Both results can be extended to support
three-dimensional five-sided reporting queries.
  Next, we turn to the four-dimensional orthogonal range reporting problem and
present a data structure that answers queries in optimal $O(\log n/\log \log n
+ k)$ time, where $k$ is the number of points in the answer. This is the first
data structure that achieves the optimal query time for this problem.
  Our results are obtained by exploiting the properties of three-dimensional
shallow cuttings.
