Cramér transform, half-space depth and threshold phenomena for convex bodies

We study the relationship between the Cramér transform and Tukey's half-space depth for log-concave probability measures. For the uniform probability measure $μ_K$ on a convex body $K\subseteq\mathbb{R}^n$, we prove the sharp pointwise comparison $$Λ_K^*(x)\leq -\log q_K(x)\leq Λ_K^*(x)+\frac12\log