Dynamic (1+ε)-approximation of the minimum-volume bounding box for moving 3D point sets

Develop an algorithm or data structure that maintains a (1+ε)-approximation of the minimum-volume bounding box for a moving set of points in three-dimensional Euclidean space, updating the approximation dynamically as the points move.

Background

This paper presents near-linear-time algorithms that compute a (1+ε)-approximation to the minimum-volume bounding box of a static set of points in R^3, along with a simpler alternative algorithm that is easier to implement but has a slightly worse asymptotic runtime.

Extending these static results to dynamic settings, where the point set moves over time, is important for applications in graphics, collision detection, and spatial data structures. The authors explicitly pose the problem of maintaining such an approximation dynamically, indicating that an efficient kinetic or dynamic solution remains open.