Let $\mathscr{P}$ be the set of row vectors of size $d$ with non-negative coefficients whose coordinate sum equals 1: $$\mathscr{P} = \left\{ u \in \mathscr{M}_{1,d}\left(\mathbb{R}_{+}\right) : \sum_{j=1}^{d} u_j = 1 \right\}.$$ We consider a square matrix $P \in \mathscr{M}_d\left(\mathbb{R}_{+}\right)$ such that for all $i \in \{1,\ldots,d\}$, $$\sum_{j=1}^{d} P_{i,j} = 1$$ We further assume that there exist $\nu \in \mathscr{P}$ and $c > 0$ such that for all $i,j \in \{1,\ldots,d\}$, $$P_{i,j} \geqslant c\nu_j.$$
Show that for all $u, v \in \mathscr{P}$, $$\|uP - vP\|_1 \leqslant (1-c)\|u - v\|_1.$$ (One may introduce $R = P - cN$ where $N = (n_{i,j})_{1 \leqslant i,j \leqslant d}$ with $n_{i,j} = \nu_j$ for all $1 \leqslant i,j \leqslant d$.)