I'm trying to solve this question in preparing for the mid-term exam:
- In question 2, I think there is a typo. IMHO, I think it should be $N_2(p)^2 \le (p,x)$ rather than $N_2(p) \le (p,x)$. Here is my reasoning:
Because $S_n$ is a nonempty closed convex subset of $\mathbb R^n$, there exists a unique orthogonal projection $\bar p$ of $0$ on $S_n$. Moreover, $\bar p$ is characterized by $\forall x \in S_n: (0-\bar p, x-\bar p) \le0$. This is equivalent to $\forall x \in S_n: (\bar p,\bar p) \le (\bar p,x)$ or $\forall x \in S_n: N_2(\bar p)^2 \le (\bar p,x)$. Because $\bar p$ is unique, we have $p= \bar p$ and thus $\forall x \in S_n: N_2(p)^2 \le (p,x)$.
Could you please confirm if my observation is correct?
- I plugged $\bar p=(1 / n, \ldots, 1 / n)$ and found that it satisfies $\forall x \in S_n: N_2(\bar p)^2 \le (\bar p,x)$. By the uniqueness of $p$, we get $p=(1 / n, \ldots, 1 / n)$.
I would like to ask how to find $p$ without knowing its value beforehand?
Thank you so much for your help!