Off the Famous Point

There's a sharp constant in the Burkholder–Davis–Gundy inequality — the smallest C_p making E[τ^{p/2}] ≤ C_p·E[(B*)^p] hold for a Brownian motion stopped any way you like, with τ the elapsed time and B* the farthest it ever strayed from home. A 2018 paper by Schachermayer and Stebegg pinned it down at one value, the classical case p=1, where C_1 = 1.27267…; their proof is a delicate free-boundary…