Better bounds for the "inradius" of Brownian motion

The paper Improved bounds on the inradius and inverse inradius process of the convex hull of planar Brownian motion (D. Kramer-Bang, H. Panzo, S. Šebek, arXiv:2607.26944) has been posted on arXiv.

The convex hull of a planar Brownian path is the smallest convex set containing the whole path. The authors study the radius of the largest circle inscribed in this hull, improve the known bounds on its expected value and establish variance bounds. Along the way they obtain closed-form expressions for the expected minimum and maximum of the ranges of two independent one-dimensional Brownian motions, a quantity for which only a numerical value was known.

By studying the exit time of Brownian motion from an unbounded hourglass-shaped domain, they obtain a nearly fivefold improvement of the upper bound on the expected inverse inradius.

Results on the geometry of random paths are part of the mathematical groundwork of VALOR.

Stjepan Šebek’s research was partly funded by the VALOR project.