5 papers
Fluctuations for diameter and perimeter of convex hulls of multiple random walks
Wojciech Cygan, Tomislav Kralj, Nikola SandriÄ +3
We study the diameter and perimeter of the convex hull generated by finitely many independent planar random walks whose increments have finite second moments. The large-time fluctu…
Asymptotics and geometric flows for a class of nonlocal curvatures
Wojciech Cygan, Tomasz Grzywny, Julia Lenczewska
We consider a family of nonlocal curvatures determined through a kernel which is symmetric and bounded from above by a radial and radially non-increasing profile satisfying an inte…
Alexandrov Theorem for nonlocal curvature
Wojciech Cygan, Tomasz Grzywny
In this article we obtain a nonlocal version of the Alexandrov Theorem which asserts that the only set with sufficiently smooth boundary and of constant nonlocal mean curvature is…
Bounds on the size of the convex hull of planar Brownian motion and related inverse processes
Wojciech Cygan, Hugo Panzo, Stjepan Å ebek
We establish bounds on expected values of various geometric quantities that describe the size of the convex hull spanned by a path of the standard planar Brownian motion. Expected…
Iterated-logarithm laws for convex hulls of random walks with drift
Wojciech Cygan, Nikola SandriÄ, Stjepan Å ebek +1
We establish laws of the iterated logarithm for intrinsic volumes of the convex hull of many-step, multidimensional random walks whose increments have two moments and a non-zero dr…