# Smallest value of largest angle in finite planar configurations

Does every set of $n$ points in the Euclidean plane contain three points $A,B,C$ such that the two segments obtained by joining $A,B$, respectively $A,C$ form an angle at least equal to $(1-2/n)\pi$ at the point $A$? (Equality is of course achieved by the vertex set of a regular $n-$gone.)

Pietro Majer's example below can be generalized and shows that $(1-2/n)\pi$ has to be replaced by a somewhat smaller constant (at least for values of $n$ which are large enough). For his example we have to take $(1-2/6)\pi$ instead of $(1-2/7)\pi$. Is the best possible constant $(1-a(n))\pi$ asymptotically substantially better, ie. can $na(n)$ become for example arbitrarily large for $n$ large enough? (It is of course obvious that $a(n)$ is decreasing but how fast?)

Update: For $n=5$, one can get arbitrarily close to $(1-1/4)\pi$: Take a right-angled isocele triangle. Split the right-angled-vertex infinitesimally along a line parallel to the longest side of the initial triangle and add an additional point on the symmetry axis very high above the two infinitesimal points.

The key bound is $$(1 - 1/n) \pi$$, due to Erdős and Szekeres: The above is an excerpt from this paper: The Erdős-Szekeres result is in their 1961 paper, "On some extremum problems in elementary geometry.", Ann. Univ. Sci. Budapest. Rolando Eötvös, Sect. Math. 3-4, 53-62 (1961) (PDF download link).

• How does the configuration of $8=2^3$ points without angles greater than $2/3\pi+\epsilon$ look like? Nov 14, 2012 at 12:44
• @Roland: I added a link to the Erdős-Szekeres paper. Their proof is in Section 4, p.59 ff. I cannot pursue it myself at the moment; sorry. Nov 14, 2012 at 13:15
• Thank you. By the way, the configuration on $8$ points is easy: one can split the central point in Pietro Majer's example into two infinitesimally close points on a line parallel to a side of the initial triangle. Nov 14, 2012 at 14:55
• actually this way we make a new angle (1−1/6)π... Nov 14, 2012 at 17:39
• I am not sure whether it's just me or whether the link is dead. In any case, this link seems to work. Here is also Wayback Machine link. Aug 15, 2019 at 6:58

Consider an exagon obtained from an equilateral triangle by cutting three small equilateral triangles from its vertices. Consider the configuration of the $6$ vertices of the exagon, plus the center of the initial equilateral triangle. It seems to me that with these $7$ points one can't do angles larger than $2\pi/3+\epsilon$.

• Indeed. So what is the best possible constant for a given integer $n$? Nov 14, 2012 at 9:14
• Yes, it seems a beautiful and difficult variational problem. Nov 14, 2012 at 10:01
• Sorry, since I can accept only one answer and since Joseph O'Rourke mention of the Erdos-Szekeres paper kills the problem completely, I have changed the accepted answer to his reply. Nov 15, 2012 at 9:50