The smallest altitude amongst the triangles formed by points in the unit circle

Question

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 $\inf(E)\ge 1$.

Answer 1

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.