Sections and projections of the outer and inner regularizations of a convex body

We establish new geometric inequalities comparing the volumes of sections and projections of a convex body, whose barycenter or Santal\'o point is at the origin, with those of its inner and outer regularizations. We also provide functional extensions of these inequalities to the setting of log-concave functions. Our approach relies on the recent optimal $M$-estimate of Bizeul and Klartag for isotropic convex bodies.