additive discrepancy under a multiplicative constraint

Question

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 $\sum_i a_i \ge 1/2 + \epsilon$.

Answer 1

$\newcommand{\ep}{\epsilon}$

It is not hard to show (see the proof at the end of this answer) that for any real $a,b,c,d\ge0$ such that $ad=bc$ we have \begin{equation}\tag{1} |a-b|+|a-c|\ge a-d. \end{equation} Replacing here $a,b,c,d$ by $a_i,b_i,c_i,d_i$ and summing in $i$, we have \begin{equation} \sum_i (|a_i - b_i| + |a_i - c_i|)\ge\sum_i a_i-\sum_i d_i\ge2\ep, \end{equation} as desired. (The conditions that $\sum_i b_i, \sum_i c_i \ge 1/2 + \ep$ were not needed or used here.)

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Proof of (1). If $a=0$, then $a-d\le0$, so that (1) holds. So, without loss of generality (wlog), $a>0$, whence $d=bc/a$ and $a-d=(a^2-bc)/a$. So, wlog $a^2>bc$ and hence $a\ge b\wedge c$. Also, wlog $c\le b$. So, one of the following two cases takes place.

Case 1: $0\le c\le b\le a$. Here (1) can be rewritten as $$f(a,b,c):=a-b-c+bc/a\ge0,$$ which follows because $f(a,b,c)$ is nonincreasing in $b$ (given that $c/a\le1$) and hence $f(a,b,c)\ge f(a,a,c)=0$. So, (1) holds in Case 1.

Case 2: $0\le c\le a\le b$. Here (1) can be rewritten as $$g(a,b,c):=b-c+bc/a-a\ge0,$$ which follows because $g(a,b,c)$ is nondecreasing in $b$ and hence $g(a,b,c)\ge g(a,a,c)=0$. So, (1) holds in Case 2 as well.

Inequality (1) is completely proved.