The isotropy constant and boundary properties of convex bodies

Let ${\mathcal{K}}^n$ be the set of all convex bodies in ${\mathbb{R}}^n$ endowed with the Hausdorff distance. We prove that if $K\in {\mathcal{K}}^n$ has positive generalized Gauss curvature at some point of its boundary, then $K$ is not a local maximizer for the isotropy constant $L_K$.