  We prove that for any semi-norm $\|\cdot\|$ 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 $K.$
  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 $f(x)=|\langle v,n_x\rangle|$, for all symmetric
convex bodies $K$, where $v\in\mathbb{R}^n$ is an arbitrary vector.
