Bounds on the quality of the PCA bounding boxes

Principal component analysis (PCA) is commonly used to compute a bounding box of a point set in Rd. The popularity of this heuristic lies in its speed, easy implementation and in the fact that usually, PCA bounding boxes quite well approximate the minimum-volume bounding boxes. We present examples of discrete points sets in the plane, showing that the worst case ratio of the volume of the PCA bounding box and the volume of the minimum-volume bounding box tends to infinity. Thus, we concentrate our attention on PCA bounding boxes for continuous sets, especially for the convex hull of a point set. Here, we contribute lower bounds on the approximation factor of PCA bounding boxes of convex sets in arbitrary dimension, and upper bounds in R2 and R3.