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Let $S$ be a finite set of points inside the unit circle. Consider all possible triangles formed by three distinct points in $S$, and among all such triangles find the smallest altitude. Denote this smallest altitude length by $A(S)$. Now let $A(n)$ denote the supremum of $A(S)$ as $S$ varies over all sets of $n$ points in the unit disc. How does $A(n)$ behave for large $n$?

Specifically let $E$ denote the set of exponents $\alpha$ such that $$ \limsup_{n\to \infty} n^{\alpha} A(n) = \infty. $$ Can one determine $\text{inf} (E)$? The original question asked whether $1.1$ belongs to the set $E$?

Two observations: By choosing $n$ evenly spaced points on the unit circle, one sees that any $\alpha >2$ is an element of $E$.

For any set $S$ of size $n$, pick a point $v$, and find two other points such that the angle formed at $v$ is $\le C/n$ for some constant $C$. Then one altitude in that triangle must be bounded by $D/n$ for some constant $D$. This shows that $\inf(E)\ge 1$.

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    $\begingroup$ Just curious --- are you also the author of mathoverflow.net/questions/142929/would-this-go-to-0/142931 ? $\endgroup$ Oct 3, 2013 at 23:27
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    $\begingroup$ I think this is an interesting question, but you should provide some more information or motivation to attract attention to your problem. Let $\alpha$ be the infimum of exponents for which there exist large sets $S$ with $n^{\alpha} a_n$ tending to infinity. Greg Martin has observed that $\alpha\le 2$. Pietro Majer has observed that $\alpha \ge 1/2$. Your original question was with exponent $1.1$. Do you have a guess for what the right exponent should be? $\endgroup$
    – Lucia
    Oct 5, 2013 at 0:44
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    $\begingroup$ To n40886: I'm sorry you don't like the edits I made, but you can simply revert back to your original version. In my opinion, to withhold relevant information that may help the reader is not such a great idea. $\endgroup$
    – Lucia
    Oct 6, 2013 at 5:48
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    $\begingroup$ "Please respect the copyright on the original question. Any reproduction, modification, use, or republication of any part of my original question without a permission is prohibited." You've come to the wrong place. First of all, anyone with enough points can edit your question to her heart's content, whether you like it or not. As for reproduction, and so on, you'd best familiarize yourself with the discussions in meta about the kind of agreement you have entered into by posting here. $\endgroup$ Oct 6, 2013 at 6:24
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    $\begingroup$ Also, I agree with @Lucia about withholding relevant information, only I would state the objection much more strongly (since I am not as polite as Lucia). $\endgroup$ Oct 6, 2013 at 6:29

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Rather than continue to edit the original question (which did not give much motivation), I'll post a few more comments here. This problem is closely related to an interesting problem of Heilbronn: given $n$ points in the unit circle consider the area of the smallest triangle formed by three of these points, and then maximize that over all choices of $n$ points. Heilbronn conjectured that this is bounded by $C/n^2$, but this was disproved by Komlos, Pintz and Szemeredi in 1980 who showed that there exist configurations with area $\ge C (\log n)/n^2$. A good summary of results on this problem may be found on the wikipedia page: http://en.wikipedia.org/wiki/Heilbronn_triangle_problem .

Since the area of a triangle in the unit circle is at most a constant times the smallest altitude, it follows that for some configuration the smallest altitude is also $\ge C(\log n)/n^2$ for some constant $C$. In other words the point $2$ actually belongs to the set $E$ defined in the problem.

For the Heilbronn problem, Roth was the first to obtain nontrivial upper bounds on the smallest area, and the best current result, due to Komlos, Pintz and Szemeredi, shows that the smallest area is always bounded by $n^{-8/7+\epsilon}$. If a number below $8/7$ were to lie in $E$, then this points to a significant difference between the altitude problem and the area problem. This seems at least plausible, since one may expect the area to be very small only if both the altitude and the base are small, but I don't know.

The Heilbronn triangle problem is also addressed in this MO question Question on a concrete example of n points , which is unfortunately poorly written.

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