Compensoids

Compensoids are a proposed class of geometric objects used to model compensation phenomena in data representation and geometric modeling. They are defined as level sets of a compensation functional that combines a base norm with a distortion-adjustment term. The base norm measures size, while the adjustment term encodes how local distortions are counteracted.

In construction, a compensoid arises by applying a compensation mapping that reweights distances according to a distortion kernel. The resulting shapes are the level sets of the resulting gauge function. In simple cases, where the distortion term is absent, compensoids reduce to ordinary ellipsoids. With directional or nonlinear adjustments, compensoids can exhibit non-elliptical, anisotropic cross-sections that reflect the nature of the distortions being counteracted.

Properties of compensoids depend on the chosen kernel. They may be convex or non-convex, and they are often star-shaped about the origin if the kernel is homogeneous and nonnegative. Some compensoids preserve certain symmetries under transformations that leave the kernel invariant. Translation generally affects the shape unless the kernel is translation-invariant, and scaling preserves the overall form up to a factor.

Applications are exploratory and interdisciplinary, spanning theoretical geometry, computer graphics, image processing, and robust statistics. The concept provides a framework for visualizing and analyzing how distortions can be offset in representations and models, without committing to a single standard definition. The term is used in contemporary discussions as a descriptive tool rather than an established, universally adopted object.

See also:

- Ellipsoid

- Gauge

- Minkowski functional

- Level set

- Norm

- Distortion

- Compensation (signal processing)