Eikonveksien

Eikonveksien is a Finnish term that appears in some mathematical contexts to denote a class of convex bodies related to the eikonal equation and distance-transform concepts. The form is the plural genitive of eikonveksi, literally implying “eikonal convexes.” In use, the exact definition of eikonveksien can vary between sources, but the idea connects convexity with an eikonal-type distance structure.

Informally, a compact set K in Euclidean space R^n is called an eikonveksi if it is convex and, with respect to a fixed eikonal distance function, the boundary of K can be described as a level set of that distance, or equivalently, if for any two points in K the shortest path measured in the eikonal metric lies entirely within K. In practice, this ties the geometry of the set to front-propagation models where the distance function d(x) satisfies |∇d| = 1 almost everywhere, and the set is compatible with this metric in a convex sense.

Key properties commonly attributed to eikonveksien include closure under intersection, and invariance under affine transformations: if K is an eikonveksi, then any affine image of K is typically regarded as an eikonveksi as well. They are not universal across all convex bodies, but standard shapes such as balls, ellipsoids, and many polyhedra often serve as examples.

Applications of the concept appear in computational geometry, robotics and motion planning, computer graphics, and medical imaging, where distance transforms and front advancing methods are used. Note that in Finnish literature the term can be idiosyncratic to particular authors or subfields, so precise definitions should be checked in a given source. See also convex set, eikonal equation, and distance transform.