Prove that $|||A||| \leq |||I|||$ for every doubly stochastic matrix $A \in M_n$. Let $|||\cdot|||$ be a unitarily invariant norm on $M_n$.
Do you (experts) agree with these proofs? Please correct me if I am making any mistakes. Thank you so much in advance.
Proof#1: Utilizing Birkhoff-Von Neumann Theorem, the doubly stochastic matrix $A$ can be described by a convex combination of permutation matrices, i.e., $$A = \sum_i \alpha_i P_i , $$ in which $\sum_i \alpha_i = 1$, $\alpha_i \geq 0 \ \forall i$, and $P_i$ are the permutation matrices (with norm $1$).
So, by utilizing the properties of the matrix norm $$|||A||| \leq \sum_i |\alpha_i| \ \underbrace{|||P_i|||}_{=|||I|||} = |||I||| \sum_i \alpha_i = |||I|||. $$
Proof#2: See herein Why does $\left\| {\left| A \right|} \right\| \le \left\| {\left| I \right|} \right\|$, for every doubly stochastic matrix $A \in M_n$?