Gaußgörbület

Gaußgörbület is a term used in German-language mathematical discussions to denote an isodensity contour of a Gaussian distribution. The concept highlights the belt-like, concentric nature of level sets around the distribution’s center. In two dimensions, it is defined as the level set of the standard Gaussian density g(x, y) = (1/(2π)) exp(- (x^2 + y^2)/2). For a fixed density level c with 0 < c ≤ 1/(2π), the Gaußgörbület L_c is the set { (x, y) : g(x, y) = c }.

Because the density is radially symmetric, each L_c is a circle centered at the origin with radius r = sqrt(-2 ln((2π) c)). In n dimensions, the Gaußgörbület is the sphere of radius r = sqrt(-2 ln((2π)^{n/2} c)). As c decreases, r increases, yielding a family of nested level sets.

Properties: these level sets are smooth, convex boundaries that are invariant under rotations. They provide a simple visualization of Gaussian mass distribution and are used as a pedagogical tool in statistics and related fields.

Examples: If c = 1/(2π) ≈ 0.159, then r = 0 (a point). If c = 0.1, r ≈ sqrt(-2 ln(0.6283)) ≈ 0.96.

Applications: teaching visualization of Gaussian densities; describing isodensity contours in statistics and image processing; serving as an intuitive aid in probabilistic geometry.

Notes: The term Gaußgörbület is not a standard formal label in most mathematical texts and is used here in a descriptive, informal sense to emphasize the belt-like structure of Gaussian level sets.

See also: Gaussian distribution, level set, isodensity contour.