Symmetry-dependence in Rounding of a Convex Body

The symmetry measure of a convex body $S\subset\mathbb{R}^n$ is given by $$\operatorname{sym}(S):=\max\{α\ge0:\text{ there exists }x\in S\text{ such that }-α(S-x)\subseteq S-x\},$$ where such an $x$ is called a Minkowski center. We prove that every convex body $S$ admits a $\sqrt{\frac{n}{\operatorn