We consider the perturbation of Schwarzschild solution by perimeter action. The asymptotic behaviour at infinity and horizon are calculated analysed in first approximation. is characterised ratio Plank length to radius. It demonstrated that modified theory appear space-time regions radius scale unreachable test particles. These located places where standard gravity has singularities.

We derive area limit laws for the various symmetry classes of staircase polygons on the square lattice, in a uniform ensemble where, for fixed perimeter, each polygon occurs with the same probability. This complements a previous study by Leroux and Rassart, where explicit expressions for the area and perimeter generating functions of these classes have been derived. MSC numbers: 82B41, 05A16, 3...

In this paper we consider an isotropic variant of the BMO-type norm recently introduced in [6]. We prove that, when considering characteristic functions of sets, this norm is related to the perimeter. A byproduct of our analysis is a new characterization of the perimeter of sets in terms of this norm, independent of the theory of distributions.

What is the minimum perimeter of a convex lattice n-gon? This question was answered by Jarník in 1926. We solve the same question, and prove a limit shape result, in the case when perimeter is measured by a (not necessarily symmetric) norm.