Consider four sequences of numbers, $0 \le a_i, b_i, c_i, d_i \le 1$, suppose they satisfy the following constraints:

(1). $\sum_{i=1}^K a_i, \sum_{i=1}^K b_i, \sum_{i=1}^K c_i \ge 1/2 + \epsilon$;

(2). $\sum_{i=1}^K d_i \le 1/2 - \epsilon$;

(3). $a_i d_i = b_i c_i$ for all $i=1, \ldots, K$.

Is it true that \begin{equation} \sum_{i=1}^K (|a_i - b_i| + |a_i - c_i|) = \Omega(\epsilon) ? \end{equation}

The following example shows that the absolute value and the sum is necessary: \begin{align*} a_1 = 1/2 + \epsilon, \quad &a_2 = \epsilon^2, \\ b_1 = 1/2 + \epsilon + \epsilon^2, \quad &b_2 = 0, \\ c_1 = 0, \quad &c_2 = 1/2 + \epsilon + \epsilon^2, \\ d_1 = 0, \quad &d_2 = 0. \end{align*}

Note that the following case is easy: if we have an explicit lower bound $b_i \ge c > 0$ for all $i$, then let $j$ be the index that maximizes $c_i/d_i$, then \begin{equation} a_j / b_j = c_j/d_j \ge (\sum_i c_i)/(\sum_i d_i) \ge 1 + \epsilon. \end{equation} Hence \begin{equation} a_i - b_i \ge c\epsilon. \end{equation} Similarly if we have a lower bound for $c_i$. Note that in these cases we don't even need the assumption that $\sum_i a_i \ge 1/2 + \epsilon$.