On local Liakopoulos-Meyer type inequalities and their functional counterparts

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 $\sigma\subset[n]$, solving a question studied by Brazitikos, Giannopoulos, Liakopoulos in [14] as well as Alonso-Guti\'errez, Bernu\'es, Brazitikos, Carbery in [3].