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Statement

Notation as on the Theorem 1 page.

Small values (p. 54). The paper says that one can easily verify

$$\alpha(3)=\tfrac13\pi,\quad\alpha(4)=\tfrac12\pi,\quad \alpha(5)=\tfrac35\pi,\quad\alpha(6)=\alpha(7)=\alpha(8)=\tfrac23\pi,$$

and that the strict inequality (3) holds for $m=7$ and $8$. For $3\le m\le6$ the regular $m$-gon is a configuration whose largest angle equals $\alpha(m)$, so (3) fails there; the authors know of no other $m$ for which equality in (1) is needed.

Proof pointer

None in the paper: the values are asserted as easy to verify. The value $\alpha(8)=\tfrac23\pi$ with the strict inequality at $m=8$ is also the case $n=3$ of Theorem 1, and $\alpha(7)=\tfrac23\pi$ the case $n=3$ of Theorem 3.

Read depth. Claims checked: the values and the remarks around them were read clause by clause on p. 54 of the print; the values were not independently verified. Nothing here is independently reviewed.

Source. P. Erdős and G. Szekeres, On some extremum problems in elementary geometry, Ann. Univ. Sci. Budapest. Eötvös Sect. Math. 3--4 (1960/1961), 53--62; the edition read is named on the source card.

Bears on

• Problem 504: the values of $\alpha_m$ for $3\le m\le8$, asserted without proof except where Theorems 1 and 3 cover them.