Convex hulls are useful as tight bounding proxies for a variety of tasks including collision detection, ray intersection, and distance computation. Unfortunately, the complexity of polyhedral convex hulls grows linearly with their input. We consider the problem of conservatively simplifying a convex hull to a specified number of half-spaces while minimizing added volume or surface area. By working in the dual representation, we propose an efficient $O(n \log n)$ greedy optimization. In comparisons, we show that existing methods either exhibit poor efficiency, tightness or safety. We demonstrate the success of our method on a variety of input shapes and downstream application domains.

翻译：凸包作为紧致包围体，在碰撞检测、射线求交和距离计算等多种任务中具有实用价值。然而，多面体凸包的复杂度随输入规模线性增长。本文研究如何在将凸包保守简化为指定数量半空间的同时，最小化新增体积或表面积。通过利用对偶表示，我们提出了一种高效的$O(n \log n)$贪心优化方法。对比实验表明，现有方法在效率、紧致性或安全性方面存在不足。我们在多种输入形状及下游应用领域验证了所提方法的有效性。