Focius

Focius is a term used in some mathematical and computational discussions to denote a central reference point for a finite set of anchors within a metric space. The focius acts as a hub that minimizes aggregate distance to the anchors. The concept is not standardized, and different authors define it in slightly different ways; in many contexts it overlaps with the geometric median or Fermat-Weber point.

Formal definition: Let anchors A = {a1, ..., an} lie in a metric space (X, d) with weights w_i > 0. A focius is a point x ∈ X that minimizes J(x) = sum w_i d(x, a_i). If x is restricted to a subset L ⊆ X, the constrained focius is the minimizer of J over L. In Euclidean space with equal weights, the unconstrained focius coincides with the geometric median; for two anchors, every point on the segment between them minimizes J.

Computation and properties: In general there is no closed-form solution. Algorithms adapted from the Fermat-Weber problem, such as iterative updates, are used to compute a focius. Uniqueness and existence depend on the space and constraint set; convexity and distance properties play key roles.

Applications: The focius serves as a robust center in facility location, clustering initialization, and network design. It provides a single representative point that balances distances to multiple anchors.

Origin and usage: The term appears in speculative and contemporary discussions rather than as a standard mathematical object; readers should verify definitions in context. See also: focus, geometric median, Fermat point, Weber problem.