# A tricky geometric inequality

Given the points $A_1,A_2,\ldots, A_n$ and $B$ and their mean $C=\frac{1}{n+1}(B+\sum_{i=1}^n A_i)$ in the Euclidean plane, what are sufficient conditions on these points such that $\sum_{i=1}^n |A_iB| \leq c\cdot(\sum_{i=1}^n |A_iC| +|BC|)$ for some coefficient $c\in \mathbb{R}$, indepedent of $n$ (The conditions may, and likely will depend on $n$)?

In particular, I'm looking for conditions that formalize the idea that most bounds eventually hold when $B$ is sufficiently close to the points $A_i$ w.r.t the distance among the $A_i$'s.

For illustration, the case $n=4$ is depicted in the figure below, where I want to bound the total length of the blue segments within a factor $c$ of the total length of the red segments.

Note that, by the triangle inequality: $$\sum_{i=1}^n |A_iB| \leq \sum_{i=1}^n (|A_iC|+|BC|)=\sum_{i=1}^n |A_iC| +n\cdot|BC| \leq n\cdot(\sum_{i=1}^n |A_iC| +|BC|),$$ but there $c$ depends clearly on $n$ (and there is some large 'slack' in the inequality, which may offer some room for improvement)

I would already be happy with conditions for the bound of $c=\frac{1}{2}$ for $n=3$ as a special case. I attempted to approach this case using the law of cosines, but I failed to distill any simple conditions from them. Additionally, I've experimentally seen that any bound can be satisfied as long as $B$ is 'close enough' to the $A_i$'s, w.r.t. the distance among the $A_i$'s, but I'm unable to formalize these conditions such that some inequality independent of $n$ holds.

• You might wish to write $$C = \frac{1}{n+1}\left( B + \sum_{i=1}^n A_i \right)$$instead, to avoid any confusion as to the definition of $C$. Dec 18, 2017 at 21:24
• Perhaps the same should be said of the inequality, writing $|BC|$ before the summation. Dec 18, 2017 at 23:27

Well, one pretty obvious sufficient condition is that $B=C$, that is,
$$\sum_{i=1}^nA_i=nB.$$
• I admit I've been a bit vague. What I'd like is some clean sufficient and nessecary conditions for some inequality to hold. However, as I doubt that such conditions are simple to find (if they even exist), I'm ok with necessary conditions that are still 'broad', in the sense that they are satisfied by a 'large' amount of initial conditions. For example, a condition such as $|\frac{1}{n}\sum_{i=1}^n A_i - B| < d$ is something I'd like to see (if it gives a bound). Dec 19, 2017 at 7:30