A new definition of affine invariant erosion of 3D surfaces is introduced. Instead of being based in terms of Euclidean distances, the volumes enclosed between the surface and its chords are used. The resulting erosion is insensitive to noise, and by construction, it is affine invariant. We prove some key properties about this erosion operation, and we propose a simple method to compute the erosion of implicit surfaces. We also discuss how the affine erosion can be used to define 3D affine invariant robust skeletons.