 Support function

The support function of a nonempty subset $C$ of a finite-dimensional real inner-product space isIt gives the largest value of a linear objective in direction $x$, allowing $+\infty$ when the set is unbounded in that direction. As a supremum of linear functions, it is convex and lower semicontinuous; it satisfies ${\sigma }_C(tx)=t{\sigma }_C(x)$ for $t>0$ and ${\sigma }_C(0)=0$.

For the Euclidean unit ball, the Cauchy-Schwarz inequality gives ${\sigma }_C(x)=\Vert x{\Vert }_2$, attained by $v=x/\Vert x{\Vert }_2$ when $x\ne 0$. For the probability simplex, the weighted-average bound gives ${\sigma }_C(x)={\max }_i{x}_i$, attained at a maximizing coordinate vector. Thus many familiar convex objectives are support functions.

Let ${\delta }_C$ be the indicator functional of a constraint set, equal to zero on $C$ and $+\infty$ outside it. The convex conjugate definition gives ${\delta }_C^{\ast }={\sigma }_C$. If $C$ is closed and convex, the Fenchel-Moreau theorem then gives ${\sigma }_C^{\ast }={\delta }_C$. This uses the extended-real constraint indicator, rather than the zero-one indicator function used in measure theory.^[ref]

The support-function subgradients as exposed faces identifies the maximizing vectors, linking sensitivity of the support function to the geometry of $C$.

The term "support function" can refer to different concepts depending on the context in which it is used. Here are a few common interpretations: 1. **Business Context**: In a business or organizational setting, support functions are departments or activities that assist the core operations of the business. Examples include human resources, IT support, customer service, and finance. These functions do not directly contribute to the production of goods or services but provide essential services that enable the core functions to operate smoothly.