Biological significance of correlation between the radius of gyration and number of layers of convex hull of protein structure
From reading the paper, I don't see any obvious biological significance. The abstract says:


  Proteins exhibit intriguing diversity in their three-dimensional conformations that enables them to perform different functions


Which is great, but there is no connection in this paper between conformation and function - not surprisingly as this is a large part of structural protein analysis!

The way I look at this work is similar to the idea of a "descriptor" from the field of Cheminformatics. The radius of gyration is a descriptor, and so is this hull number. The author claims a correlation between these two descriptors, so... is one easier to calculate than the other?

This is interesting work, but I'm not sure at this point what the significance of hull number is for protein structural analysis. However I can say that the correlation between these two descriptors is not biologically significant - it's just a geometric relationship that would presumably hold for any set of 3D points.