Inequalities for sections and projections of log-concave functions

We provide extensions of geometric inequalities about sections and projections of convex bodies to the setting of integrable log-concave functions. Namely, we consider suitable generalizations of the affine and dual affine quermassintegrals of a log-concave function $f$ and obtain upper and lower estimates for them in terms of the integral $|f|_1$ of $f$, we give estimates for sections and projections of log-concave functions in the spirit of the lower dimensional Busemann-Petty and Shephard problem, and we extend to log-concave functions the affirmative answer to a variant of the Busemann-Petty and Shephard problems, proposed by V. Milman. The main goal of this article is to show that the assumption of log-concavity leads to inequalities in which the constants are of the same order as that of the constants in the original corresponding geometric inequalities.

💡 Research Summary

The paper “Inequalities for sections and projections of log‑concave functions” extends a number of classical geometric inequalities concerning sections and projections of convex bodies to the functional setting of integrable log‑concave functions. The authors introduce functional analogues of the affine and dual‑affine quermassintegrals, provide sharp upper and lower bounds for these quantities in terms of the L¹‑norm of the function, and solve functional versions of Milman’s variant of the Busemann‑Petty and Shephard problems, including lower‑dimensional analogues.

Key definitions: for a log‑concave function f : ℝⁿ→