  U-max statistics were introduced by Lao and Mayer in 2008. Instead of
averaging the kernel over all possible subsets of the original sample, they
considered the maximum of the kernel. Such statistics are natural in stochastic
geometry. Examples are the maximal perimeters and areas of polygons and
polyhedra formed by random points on a circle, ellipse, etc. The main method to
study limit theorems for U-max statistics is a Poisson approximation. In this
paper we consider a general class of kernels defined on a circle, and we prove
a universal limit theorem with the Weibull distribution as a limit. Its
parameters depend on the degree of the kernel, the structure of its points of
maximum and the Hessians of the kernel at these points. Almost all limit
theorems known so far may be obtained as simple special cases of our general
theorem. We also consider several new examples. Moreover, we consider not only
the uniform distribution of points but also almost arbitrary distribution on a
circle satisfying mild additional conditions.
