Proximum

Proximum is a term used in geometry and optimization to denote a point in a given set that is closest to a specified external point. More formally, let (X, d) be a metric space and A ⊆ X be nonempty. For x ∈ X, a point p ∈ A is a proximum of x in A if d(x, p) = inf{d(x, a) : a ∈ A}. If the infimum is actually attained, p is called a proximal point; the collection of all such points is the proximal set Prox_A(x).

Existence and uniqueness: In general, a proximum may fail to exist if the distance infimum is not attained. In many common spaces, existence is guaranteed under additional assumptions. For instance, in a finite-dimensional Euclidean space, a nonempty compact set A yields a proximum for any x. In Hilbert spaces, if A is nonempty closed and convex, every x has a unique proximal point in A; this point is the metric projection of x onto A. More broadly, in metric spaces with suitable compactness or convexity conditions, proximal points exist and may be unique.

Examples: If A is a circle in the plane and x lies outside A, the proximum is the point on the circle along the line from the circle’s center toward x. If x ∈ A, then x itself is a proximum. If A is a subspace or affine subspace with appropriate properties, the proximum corresponds to the usual foot of the perpendicular from x to A.

Relation to projection: The proximum concept generalizes the idea of a projection onto a set in metric spaces. The projection operator P_A(x) yields a proximal point when it exists, and in many common settings (notably closed convex sets in Hilbert spaces) the projection is unique. See also distance function, metric projection, and proximal methods in optimization.