The logarithmic Minkowski inequality for cylinders

In this paper, we prove that if $K$ is an $o$-symmetric cylinder and $L$ is an $o$-symmetric convex body in ${\mathbb{R}}^3$, then the logarithmic Minkowski inequality $$\frac{1}{V(K)}{\int }_{{\mathbb{S}}^2}\log \frac{h_L}{h_K}d{V}_K\ge \frac{1}{3}\log \frac{V(L)}{V(K)}$$ holds, with equality if and only if $K$ and $L$ are relative cylinders.