More on logarithmic sums of convex bodies

We prove that the log-Brunn-Minkowski inequality (log-BMI) for the Lebesque measure in dimension $n$ would imply the log-BMI and, therefore, the B-conjecture for any log-concave density in dimension $n$. As a consequence, we prove the log-BMI and the B-conjecture for any log-concave density, in the plane. Moreover, we prove that the log-BMI reduces to the following: For each dimension $n$, there is a density $f_n$, which satisfies an integrability assumption, so that the log-BMI holds for parallelepipeds with parallel facets, for the density $f_n$. As byproduct of our methods, we study possible log-concavity of the function $t\mapsto |(K+_p\cdot e^tL)^{\circ}|$, where $p\geq 1$ and $K$, $L$ are symmetric convex bodies, which we are able to prove in some instances and as a further application, we confirm the variance conjecture in a special class of convex bodies. Finally, we establish a non-trivial dual form of the log-BMI.