In what concerns the affine component of Felix Klein’s Erlangen program, the first results were due to Blaschke’s school. They were of differential geometric nature and required at least C^4 regularity of closed convex hypersurfaces with, often, positive Gauss curvature everywhere. While this is unsatisfactory for the general study of convex bodies, certain objects – the most famous one being the affine surface area – have appeared in affine differential geometry but they were later extended to arbitrary convex bodies with suprising applications. We want to motivate a certain direction of research which seeks new affine differential invariants and their applications in view of possible extensions to arbitrary convex bodies. The talk will not require any prerequisite.