Discrete log-concavity and threshold phenomena for atomic measures

We investigate threshold phenomena for random polytopes $K_N=\text{conv}\{X_1,\dots ,X_N\}$ generated by i.i.d.\ samples from an atomic law $\mu$. We identify and provide a missing justification in the discrete-hypercube threshold argument of Dyer–Füredi–McDiarmid, where the supporting half-space estimate is derived via a smooth (gradient/uniqueness) step that can fail at boundary contact points. We then compare threshold-driving mechanisms in the continuous log-concave setting – through the Cramér transform and Tukey’s half-space depth – with their discrete analogues. Within this framework, we establish a sharp threshold for lattice $p$-balls ${\mathbb{Z}}^n\cap r{B}_p^n$. Finally, we present structural counterexamples showing that sharp thresholds need not hold in general discrete log-concave settings.