On the local version of the Log-Brunn-Minkowski conjecture and some new related geometric inequalities

We prove that for any semi-norm $\mathrm{\parallel }\cdot \mathrm{\parallel }$ on ${\mathbb{R}}^n,$ and any symmetric convex body $K$ in ${\mathbb{R}}^n,$ \begin{equation}\label{ineq-abs2} \int_{\partial K} \frac{|n_x|^2}{\langle x,n_x\rangle}\leq \frac{1}{|K|}\left(\int_{\partial K} |n_x| \right)^2, \end{equation} and characterize the equality cases of this new inequality. The above would also follow from the Log-Brunn-Minkowski conjecture, if the latter was proven, and we believe that it may be of independent interest. We, furthermore, obtain an improvement of this inequality in some cases, involving the Poincare constant of $K\mathrm{.}$ The conjectured Log-Brunn-Minkowski inequality is a strengthening of the Brunn-Minkowski inequality in the partial case of symmetric convex bodies, equivalent to the validity of the following statement: for all symmetric convex smooth sets $K$ in ${\mathbb{R}}^n$ and all smooth even $f:\partial K\rightarrow $\mathbb{R}$,$ \begin{equation}\label{ineq-abs} \int_{\partial K} H_x f^2-\langle \mbox{{II}}^{-1}\nabla_{\partial K} f, \nabla_{\partial K} f\rangle +\frac{f^2}{\langle x,n_x\rangle}\leq \frac{1}{|K|}\left(\int_{\partial K} f \right)^2. \end{equation} In this note, we verify the above with the particular choice of speed function $f(x)=\mathrm{\mid }⟨v,n_x⟩\mathrm{\mid }$, for all symmetric convex bodies $K$, where $v\in {\mathbb{R}}^n$ is an arbitrary vector.