John property of anisotropic minimal surfaces

For a convex set K⊂ Rⁿ and the associated anisotropic perimeter PK, we establish that every (ε,\,r)-minimizer for PK satisfies a local John property. Furthermore, we prove that a certain class of John domains, including (ε,\,r)-minimizers close to K, admits a trace inequality. As a consequence, we provide a more concrete proof for a crucial step in the quantitative Wulff inequality, thereby complementing the seminal work of Figalli, Maggi, and Pratelli.