My question may be driven by luck of appropriate knowledge. So I'm sorry if I ask something which maybe obvious.
Generally support function are defined as $s_{A}(x) = \sup\{x\cdot a| a\in A\}$, for $x\in R^{n}$, and $A \subset R^{n}$. (or in some cases with more general <,> scalar product instead of dot.). Mostly it is taken $||x||=1$ (i.e. $x\in S^{n}$, S for sphere).
Mostly support function are used to uniquely characterize closed convex bodies. Meaning that if we have the support function of convex body, we can retrieve it. So we have body itself. (In this case support function is convex function.) However by one can calculate it for closed non-convex figures. So here is my question. Suppose that I have some support function for some body.
1) Can I retrieve the non-convex body with its $s_{B}(x)$ ,$x\in S^{n}$?
(I understand that for most cases it is impossible, meaning non-uniqueness, without additional conditions).
And the main one
2) Can I retrieve convex hull of some closed body $B$ uniquelly, having $s_{B}(x)$? If so, any useful techniques?
Thank you very much.
Define the indicator function: $$\delta(a) \begin{cases}0 & \text{if } a \in A \\ \infty & \text{else}\end{cases}$$ The support function is the conjugate of the indicator function (and vice versa if $A$ is convex and closed). Therefore, given the support function, you can compute the indicator function of $A$ as: $$\delta(a) = \sup\{x\cdot a - s_A(x)\}$$ Now if a function is not closed convex, taking the double conjugate results in the closed convex hull of a function (or of the epigraph of that function). So the answers to your question are: (1) no, the nonconvex part is lost (take for example the sets $[-2,2]$ and $[-2,-1] \cup [1,2]$ to see that they have the same support function), (2) yes, simply take the conjugate of the support function to obtain the indicator function of $B$.