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We prove that for any semi-norm   $\|\cdot \|$   on   $\mathbb{R}^n$   and any symmetric convex body   $K$   in   $\mathbb{R}^n,$(1)  $$\begin{align}&amp; \int_{\partial K} \frac{\|n_x\|^2}{\langle x,n_x\rangle}\leq \frac{1}{|K|}\left(\int_{\partial K} \|n_x\| \right)^2, \end{align}$$  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   $K.$

On the Local Version of the Log-Brunn–Minkowski Conjecture and Some New Related Geometric Inequalities