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Statement

Notation (printed p. 254): for a planar convex body $K$, $N=N(K)\in\mathbb N\cup{\infty}$ is the smallest number for which there is a point $P\in\partial K$ such that every circle with centre $P$ meets $\partial K$ in at most $N$ points. For $n\in\mathbb N\cup\{\infty\}$, $J(K,n)$ is the set of points $P\in\partial K$ for which some circle centred at $P$ meets $\partial K$ in at least $n$ points; by Theorem 1.2, $N(K)$ is the largest $N$ with $J(K,N)=\partial K$.

Theorem 1.1 (printed p. 254). "There is a planar convex body $K$ with $N(K)=6$."

In this notation Erdős's 1946 statement, quoted by the paper on p. 253, is $N(K)\le2$ for every convex body $K$. The paper remarks (p. 254) that it fails for every acute triangle, where each boundary point is the centre of a circle meeting the boundary 4 times, and for every regular $(2k+1)$-gon; it conjectures that $N(K)$ is bounded by a constant independent of $K$, "probably by 6" (p. 254).

Source. I. Bárány and E. Roldán-Pensado, A question from a famous paper of Erdős, Discrete Comput. Geom. 50 (2013), 253--261, doi:10.1007/s00454-013-9507-z; the definitions and Theorem 1.1 on printed p. 254. The edition read is identified on the source card.

Read depth. Claims checked: the definitions and the theorem were read clause by clause on the page image. The proof (p. 259) was read for structure only; its "direct computation" was not repeated, and nothing here is independently reviewed.

Proof pointer

§ 3, p. 259, using Lemmas 3.1 and 3.2 (p. 258). The body is a 15-gon with threefold rotational symmetry, built from the points $A_1=(1000,0)$, $A_2=(906,114)$, $A_3=(645,359)$, $A_4=(-498,871)$ and their rotations by $2\pi/3$ and $4\pi/3$ about the origin, with a fifth point $A_5$ near $A_4$ (and its rotations) supplied by Lemma 3.1 at the acute angle $\angle A_3A_4B_1$. Lemma 3.2 places, for points near the broken line $A_4B_1B_2B_3$, a centred circle meeting the broken line $C_1C_2C_3C_4$ in at least 6 points, checked by direct computation; Lemma 3.1 handles the rest of the side $[A_3,A_4]$. The paper states that the midpoint of $[A_3A_4]$ is not in $J(K,7)$, which gives $N(K)\le6$. Not checked here.

Dependencies

Lemmas 3.1 and 3.2 (p. 258) of the paper.

Bears on

• Problem 982: Erdős's 1946 paper poses three conjectures on p. 248, each stated as stronger than the one before; the problem's vertex bound is the consequence it draws from the second, and the convex-curve statement is the third. The theorem shows that the number 2 in the convex-curve statement cannot be replaced by anything below 6. It is a statement about points of a convex curve and their centred circles, says nothing about distinct distances from a vertex of a convex polygon, and leaves the problem's statement undecided.