Minimization of a linear functional over a set by support function

Question

My question comes from the famous book: convex analysis, Rockafellar Ch. 13

My question is the minimization over $C$ should be $-\delta^*(x^* \mid C)$. Why is the answer $-\delta^*(-x^* \mid C)$?

My point is that if we I get $\delta^*(x^* \mid C)$, the maximum value of $x^*$ over $C$, the minimum is nothing but the minus sign of it.

Note: A related question just for reference:

"Support function of a set" and supremum question.

Answer 1

Note that $\inf_{x \in S} \, f(x) = - \sup_{x \in S} -f(x)$.

The smallest possible value of $f$ is the same as the largest possible value of $-f$, except the signs are flipped.

So the formula $-\delta^*(-x^* | C)$ is correct.