Conjecture on the optimal dimension dependence in quantitative isoperimetric inequalities

Let $n\geq 2$, and let $C(K,n)$ and $C(n)$ denote the optimal constants in the quantitative anisotropic isoperimetric inequality and quantitative Brunn–Minkowski inequality, respectively, for compact convex bodies $K,L\subset\mathbb R^n$ of positive volume. Optimal-constant conjecture. In both quantitative inequalities, the optimal constants $C(K,n)$ and $C(n)$ are of the form $Cn^2$, where $C$ is an absolute constant. The paper proves the lower bound of order $n^2$ for the Brunn–Minkowski constant, while the matching upper bound is left as a task for future work.