Minimal boundary regularity needed for finite flat-shadow characterization

Determine how far the boundary smoothness assumptions can be reduced for theorems asserting that a convex body K in R^n is an ellipsoid when finitely many flat shadow boundaries pass through each boundary point. In particular, ascertain the lowest regularity of ∂K under which the finite-source flat-shadow hypothesis still forces K to be an ellipsoid.

Background

The main results use C^3 (and in 3D, C^5) smoothness, and the authors explain that by invoking a revised result on third-order contact they can relax the assumption for one theorem to C^{2,1}. They explicitly note that the optimal regularity threshold remains unknown and highlight interest in reducing smoothness further.