Borelveelhoek

Borelveelhoek, or Borel polygon, is a polygon in the plane defined by a finite ordered sequence of vertices that lie in a Borel subset of the plane. The edges are straight line segments connecting consecutive vertices in a cyclic order, forming a simple polygon when the edges do not cross.

In practice, a Borelveelhoek is specified by a finite set of points v1, v2, ..., vn chosen from a Borel subset B ⊆ R^2, provided these points form a simple polygon when connected in the chosen order. Because a finite subset of the plane is a Borel set, every ordinary polygon can be regarded as a Borelveelhoek. This observation means the concept is primarily of interest when considering questions about measurability, probability, or uncertainty in geometric contexts.

Properties of Borelveelhoeken include standard geometric measures such as area and perimeter, computable via the shoelace formula and edge-length sums, respectively. If the vertices are regarded as random samples from a Borel set with an associated probability measure, one can study distributions of area, shape, or other features of the resulting polygons. The notion also supports considering families of polygons with vertices constrained to a bounded Borel set, yielding bounded Borelveelhoeken.

Variants may specify additional regularity, such as requiring no three consecutive vertices to be collinear, or restricting the sampling mechanism for the vertices. In most mathematical discussions, the term serves to emphasize measurability or probabilistic aspects rather than introducing a separate, distinct geometric class beyond ordinary polygons.

See also: Borel set, polygon, measure theory, geometric probability.