Consider a convex polyhedral robot $B$ that can translate (without rotating) amidst a finite set of non-moving polyhedral obstacles in $\mathbb R^3$. The "free space" $\mathcal F$ of $B$ is the set of all positions in which $B$ is disjoint from the interior of every obstacle. Aronov and Sharir (1997) derived an upper bound of $O(n^2\log n)$ for the combinatorial complexity of $\mathcal F$, where $n$ is the total number of vertices of the obstacles, and the complexity of $B$ is assumed constant. Halperin and Yap (1993) showed that, if $B$ is either a box or a "flat" convex polygon, then a tighter bound of $O(n^2α(n))$ holds. Here $α(n)$ is the inverse Ackermann function. In this paper we prove that if $B$ is a box, then the complexity of $\mathcal F$ is $O(n^2)$. Furthermore, if $B$ is a convex polygon whose edges come in parallel pairs, then the complexity of $\mathcal F$ is $O(n^2)$ as well. These results settle the question of the asymptotical worst-case complexity of $\mathcal F$ for a box, as well as for all convex polygons.

翻译：考虑一个凸多面体机器人$B$，它能在三维空间$\mathbb R^3$中平移（不旋转）运动，环境中存在有限个静止的多面体障碍物。$B$的“自由空间”$\mathcal F$是指所有使$B$与每个障碍物内部均不相交的位置集合。Aronov与Sharir（1997）推导出$\mathcal F$的组合复杂度上界为$O(n^2\log n)$，其中$n$为障碍物的总顶点数，且假设$B$的复杂度为常数。Halperin与Yap（1993）证明，若$B$是一个盒状体或“扁平”凸多边形，则存在更紧的上界$O(n^2α(n))$，此处$α(n)$为反阿克曼函数。本文证明：当$B$为盒状体时，$\mathcal F$的复杂度为$O(n^2)$；进一步地，若$B$是一个各边成对平行的凸多边形，则$\mathcal F$的复杂度亦为$O(n^2)$。这些结果彻底解决了盒状体及所有凸多边形情形下$\mathcal F$的渐近最坏情况复杂度问题。