A new proof of the Wulff-Gage isoperimetric inequality and its applications

A new proof of the Wulff-Gage isoperimetric inequality for origin-symmetric convex bodies is provided. As its applications, we prove the uniqueness of log-Minkowski problem and a new proof of the log-Minkowski inequality of curvature entropy for origin-symmetric convex bodies of $C^2$ boundaries in ${\mathbb{R}}^2$ is given.