Sections and projections of the outer and inner regularizations of a convex body
Natalia Tziotziou
TL;DR
This work establishes sharp geometric inequalities comparing volumes of sections and projections of a convex body $K$ with those of its inner and outer regularizations, under the origin-centered barycenter or Santaló-point condition. The authors leverage isotropic normalization and the modern $MM^{\ast}$-type estimates for isotropic convex bodies to derive nearly linear dependencies on the dimension ratio $n/k$, via bounds of the form ${\rm vrad}(P_H(K_{\rm out})) \le \frac{n\,g(n)}{k}\, {\rm vrad}(P_H(K_{\rm in}))$ and ${\rm vrad}(K_{\rm out} \cap H) \le \frac{n\,g(n)}{k}\, {\rm vrad}(K_{\rm in} \cap H)$ with $g(n) \le C(\ln n)^3$. The approach also yields random-subspace (isotropic) results with high probability and extends to the functional setting via log-concave functions by employing Ball’s $K_p(f)$-construction and related auxiliary lemmas. Functional analogues of the projection and section inequalities are established for $f \in \mathcal{L}^n$, using $\Delta_{ m out}$ and $\Delta_{ m in}$ to mirror the geometric difference body. Overall, the work strengthens Blaschke–Santaló type relations for non-symmetric bodies and provides a robust framework for functional extensions in the log-concave setting, with explicit dependence on dimension through $(\ln n)$ factors.
Abstract
We establish new geometric inequalities comparing the volumes of sections and projections of a convex body, whose barycenter or Santaló 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.