Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On local Liakopoulos-Meyer type inequalities and their functional counterparts

Luis J. Alías, Bernardo González Merino, Beatriz Marín Gimeno

TL;DR

The paper extends local geometric inequalities to a functional setting by proving a Rogers–Shephard type inequality for log-concave functions on $\mathbb{R}^n$ under a $1$-reducible $s$-cover. This functional result yields sharp local Liakopoulos–Meyer type inequalities for $n$-dimensional convex bodies and solves open questions in the local Bollobás–Thomason framework. The authors develop a Brunn concavity–based approach and introduce a supremal convolution operator to bridge functionals and projections/sections, with equality cases precisely characterized. Additional improvements are obtained in the unconditional setting, including refined Alonso–Meyer bounds and detailed comparisons to prior constants, along with an appendix containing technical estimates and proofs to support the main theorems.

Abstract

We provide a functional Rogers-Shephard type inequality for log-concave functions on $\mathbb R^n$ and any $1$-reducible $s$-cover of $[n]$. As a consequence, we derive a sharp local Liakopoulos-Meyer type inequality for $n$-dimensional convex bodies and $1$-reducible $s$-covers of any $σ\subset[n]$, solving a question studied by Brazitikos, Giannopoulos, Liakopoulos in [14] as well as Alonso-Gutiérrez, Bernués, Brazitikos, Carbery in [3].

On local Liakopoulos-Meyer type inequalities and their functional counterparts

TL;DR

The paper extends local geometric inequalities to a functional setting by proving a Rogers–Shephard type inequality for log-concave functions on under a -reducible -cover. This functional result yields sharp local Liakopoulos–Meyer type inequalities for -dimensional convex bodies and solves open questions in the local Bollobás–Thomason framework. The authors develop a Brunn concavity–based approach and introduce a supremal convolution operator to bridge functionals and projections/sections, with equality cases precisely characterized. Additional improvements are obtained in the unconditional setting, including refined Alonso–Meyer bounds and detailed comparisons to prior constants, along with an appendix containing technical estimates and proofs to support the main theorems.

Abstract

We provide a functional Rogers-Shephard type inequality for log-concave functions on and any -reducible -cover of . As a consequence, we derive a sharp local Liakopoulos-Meyer type inequality for -dimensional convex bodies and -reducible -covers of any , solving a question studied by Brazitikos, Giannopoulos, Liakopoulos in [14] as well as Alonso-Gutiérrez, Bernués, Brazitikos, Carbery in [3].

Paper Structure

This paper contains 5 sections, 12 theorems, 113 equations.

Table of Contents

  1. Introduction
  2. Proofs of the inequalities
  3. Proofs of the equality cases
  4. Improved bounds in other cases
  5. Appendix

Key Result

Theorem 1.1

Let $f\in\mathcal{F}(\mathbb R^n)$ and let $(\sigma_1,\dots,\sigma_m)$ a $1$-reducible $s$-cover of $[n]$. Then, Moreover, equality holds above if and only if $f=\|f\|_\infty\chi_C$ where with $0\in K_j\in\mathcal{K}^{|\overline{\sigma}_j|}$ for $j=1,\dots,k$ and with $(\overline{\sigma}_1,\dots,\overline{\sigma}_k)$ being the $1$-cover of $[n]$ induced by the $s$-cover $(\sigma_1,\dots,\sigma_m

Theorems & Definitions (28)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Conjecture 1.4
  • Lemma 2.1
  • proof
  • Proposition 2.2
  • proof
  • Lemma 2.3
  • proof
  • ...and 18 more