Suppose we have a vector $e \in \mathbb{R}^m$, and two coordinate systems with orthonormal bases $\{v_1,v_2,\ldots,v_r\}$ and $\{w_1,w_2,\ldots,w_k\}$, $k,r\leq m$. Suppose
$$e=\sum_{i=1}^r a_iv_i = \sum_{j=1}^k b_jw_j $$
and that $a_i,b_j>0$, $a_1=\max a_i$, $b_1=\max b_j$ and $\langle v_i,w_j \rangle\geq 0$, $\forall i \in \{1,\ldots, r\}$,$\forall j \in \{1,\ldots, k\}$.
Find a lower bound for $\langle v_1,w_1 \rangle$.
Please share some ideas for the problem!
Many thanks!