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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.

The configuration for $n=4$

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.

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    $\begingroup$ 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 $C$. $\endgroup$ Dec 18, 2017 at 21:24
  • $\begingroup$ Perhaps the same should be said of the inequality, writing $|BC|$ before the summation. $\endgroup$ Dec 18, 2017 at 23:27

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Well, one pretty obvious sufficient condition is that $B=C$, that is,

$$\sum_{i=1}^nA_i=nB.$$

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  • $\begingroup$ Indeed, this condition gives equality, although this isn't quite the type of condition I'm looking for. I'll phrase more clearly what I'm looking for. $\endgroup$ Dec 18, 2017 at 22:33
  • $\begingroup$ I don't really understand your rephrasing. $\endgroup$ Dec 18, 2017 at 23:26
  • $\begingroup$ 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). $\endgroup$ Dec 19, 2017 at 7:30
  • $\begingroup$ Do you want a sufficient condition or a necessary condition? Those are very two different things. $\endgroup$ Dec 19, 2017 at 13:39
  • $\begingroup$ Well both, preferably. But as I don't think that leads to 'nice' enough conditions, sufficient and 'almost necessary' (in the sense that the set of points for which the bound holds but the conditions don't is 'small') conditions are fine too. $\endgroup$ Dec 19, 2017 at 14:14

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