SteinerSymmetrisierung

SteinerSymmetrisierung is a term that refers to a mathematical operation or concept within the field of geometry, particularly related to convex sets. It is named after the Swiss mathematician Jakob Steiner. The core idea involves transforming a given geometric object, typically a convex body, into a more symmetric form while preserving certain properties.

The process often involves averaging the object over its symmetries or related geometric transformations. For instance, if one considers a convex set in a Euclidean space, the Steiner symmetrisation might involve averaging the set with its reflections across various hyperplanes or rotationally averaging it. A common form of Steiner symmetrisation is the Steiner point symmetrization, which can be applied to convex polygons. This process involves reflecting the polygon across lines passing through a chosen point and then taking the convex hull of the original polygon and its reflections.

The resulting symmetrized object often possesses a higher degree of symmetry than the original. For example, if one starts with a convex body and applies Steiner symmetrisation with respect to all possible directions, the resulting object is a ball (or a disk in 2D). A key property preserved by Steiner symmetrisation is usually convexity. Other geometric quantities, such as volume or surface area, might change, but bounds on these quantities can often be established. This operation finds applications in various areas of convex geometry, functional analysis, and geometric measure theory, particularly in proving inequalities and understanding the geometric properties of sets.