sublevelset

A sublevel set, or sublevelset, of a function f defined on a set X is the collection of points where the function value does not exceed a given threshold. Formally, for a real-valued function f: X → R and a real number α, the sublevel set is Lα(f) = { x ∈ X : f(x) ≤ α }. It is also called a lower level set. The family of sublevel sets as α varies forms a filtration: if α ≤ β then Lα(f) ⊆ Lβ(f).

If X is a topological space and f is continuous, each sublevel set Lα(f) is closed, since it is the preimage of the closed interval (−∞, α] under a continuous map. In a vector space, the convexity of sublevel sets follows from the convexity of f: if f is convex, then for every α, Lα(f) is a convex set. More generally, quasi-convex functions are characterized by the property that all their sublevel sets are convex.

Common examples illustrate the concept. For f(x) = ||x||^2 on R^n, the sublevel set Lα(f) is the closed ball { x ∈ R^n : ||x|| ≤ sqrt(α) }. If f(x) = max(|x1|, |x2|), then Lα(f) is the closed square centered at the origin with side length 2√α.

Applications of sublevel sets arise in optimization, where they define feasible regions { x : f(x) ≤ α } and help analyze algorithmic behavior and convergence. They also appear in dynamical systems and topology, such as in Morse theory, where tracking how sublevel sets change as α passes critical values reveals information about the underlying space. See also level set and epigraph.