Support function

The support function of a set records the supremum of each linear functional over that set. It encodes convex geometry through directional bounds.

Convex conjugate: The transform that assigns a function the supremum of its pairing with a variable minus that function’s value. The indicator function’s convex conjugate is the support function.
Linear functional: A linear map from a vector space to its scalar field. Each input direction specifies the linear functional whose supremum is recorded.
Convex optimization: Optimization of a convex objective over a convex feasible set. Support functions express constraints and appear in dual optimization formulations.
Gauge function: A function measuring the smallest nonnegative scaling of a set needed to contain a point. A gauge measures points relative to a set; a support function measures sets in directions.
Indicator function: An extended-real-valued function equal to zero on a set and infinity outside it. Taking its convex conjugate produces the set’s support function.
Supremum: The least upper bound of a set of real numbers. The defining value may be a maximum, or only a supremum.
Proximal operator: An operator that minimizes a function plus a scaled squared-distance penalty. Support functions have proximal operators linked to projection onto polar sets.
Minkowski functional: The scaling function associated with a set containing the origin. It is a primal point-to-set measure, unlike the dual directional bound of a support function.
Sublinear function: A positively homogeneous, subadditive function on a vector space. Support functions obey these two defining properties.
Convex set: A set containing the line segment between every pair of its points. For closed convex sets, the support function captures the set completely.

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