Nichtglatte

Nichtglatte is a German term used in mathematics and related disciplines to describe objects that lack smoothness in a precise sense. In analysis and differential geometry, “glatt” generally denotes a function, surface, or manifold of high regularity, often of class C∞ (infinitely differentiable). A object or function is nichtglatt if it fails to meet the required smoothness criteria at some points or with respect to a chosen order of differentiability.

In practice, nichtglatte functions are those that are not differentiable at certain points or not differentiable of a given order. Common examples include the absolute value function |x|, which is not differentiable at x = 0, and the maximum function max(x, y), which is not differentiable on the line where x = y. More subtle examples include functions like x1/3, which are not differentiable at 0. In geometry, a polyhedral surface is nichtglatt along edges and vertices where the surface has corners or creases, in contrast to smooth (glatte) surfaces with well-defined tangent planes everywhere.

In optimization and numerical analysis, nichtglatte (non-smooth) problems are widespread. They require specialized techniques such as subgradient methods, subdifferentials, proximal operators, and bundle methods. Smoothing or regularization approaches may be employed to approximate nichtglatte problems by smooth ones.

Overall, nichtglatte describes a broad class of objects that do not possess the strongest forms of smoothness, with specific meaning dependent on the chosen smoothness framework (differentiability, continuity, or analytic properties).