Support function
= Support function
{title2=$\sigma_C$}
{wiki}
The <support function> of a nonempty subset $C$ of a finite-dimensional real inner-product space is
$$
\sigma_C(x)=\sup_{v\in C}\langle x,v\rangle.
$$
It 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 $\sigma_C(x)=\|x\|_2$, attained by $v=x/\|x\|_2$ when $x\ne0$. 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 $\delta_C^*=\sigma_C$. If $C$ is closed and convex, the <Fenchel-Moreau theorem> then gives $\sigma_C^*=\delta_C$. This uses the extended-real constraint indicator, rather than the zero-one <indicator function> used in measure theory.<Convex Optimization by Stephen Boyd and Lieven Vandenberghe>[§3.3.1, Example 3.24, conjugate of a set indicator]{ref}
The <support-function subgradients as exposed faces> identifies the maximizing vectors, linking sensitivity of the support function to the geometry of $C$.