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Abstract

Let $S$ be a set of $n$ points in $d$-space, let $R$ be an axes-parallel hyper-rectangle and let $1 \leq k \leq n$ be an integer. An algorithm is given that decides if $R$ can be translated such that it contains at least $k$ points of $S$. After a presorting step, this algorithm runs in $O(n)$ time, with a constant factor that is doubly-exponential in~$d$. Two applications are given. First, a translate of $R$ containing the maximal number of points can be computed in $O(n \log n)$ time. Second, a $k$-point subset of $S$ with minimal $L_{\infty}$-diameter can be computed, also in $O(n \log n)$ time. Using known dynamization techniques, the latter result gives improved dynamic data structures for maintaining such a $k$-point subset.