Asymmetry of convex sets with isolated extreme points

When measuring asymmetry of convex sets $\mathcal{L}\subset\br^n$ in terms of inscribed simplices, the interior of $\mathcal{L}$ naturally splits into regular and singular sets. Based on examples, it may be conjectured that the singular set is empty iff $\mathcal{L}$ is a simplex. In this paper we prove this conjecture with the additional assumption that $\mathcal{L}$ has at least $n$ isolated extreme points on its boundary.