Compute the support function, $\sigma_{C}$, when $C$ is a subspace.

Question

Recall that in convex analysis the support function is the conjugate of the indicator function and is defined to be $\sigma_{C}(x)=\sup_{v\in C} \langle v,x\rangle$. Compute the support function, $\sigma_{C}$, when C is a subspace.

Please help me with starting this proof. Thank you.

Answer 1

If $x \in C^\perp$, then $x^Tv = 0$ for all $v \in C$ and so $i_C^*(x) = 0$. If $x \not \in C^\perp$, then $x^Tv_0 \not = 0$ for some $v_0 \in C\setminus\{0\}$. Now, it is clear that $v \mapsto x^Tv$ is unbounded on the line $L := \{tv_0 | t \in \mathbb{R}\} \subseteq C$. Thus $i_C^*(x) = +\infty$. Putting things together, we have $i_C^* = i_{C^\perp}$, as claimed.

Answer 2

$\sigma_C=\iota_{C^\perp}$ where $C^\perp$ is the orthogonal of $C$ and $\iota_K$ is the indicator function of $K$