centerpreserving

centerpreserving refers to a type of map or transformation in mathematics that preserves the midpoint of line segments. A map f is centerpreserving if for any two points x and y, the image of the midpoint of the segment connecting x and y is the midpoint of the segment connecting the images of x and y. Mathematically, this can be expressed as f((x+y)/2) = (f(x)+f(y))/2 for all x and y in the domain of f.

This property is closely related to linearity. In vector spaces, linear transformations are always centerpreserving. However, not all centerpreserving maps are necessarily linear. For instance, certain affine transformations are centerpreserving. The concept is particularly relevant in geometry, functional analysis, and the study of convex sets. Understanding centerpreserving maps can provide insights into the structure of spaces and the behavior of transformations within them. The condition implies that the map acts uniformly across the space with respect to midpoints.