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Statement

Notation (printed p. 254): $N(K)\in\mathbb N\cup\{\infty\}$ 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 $N(K)$ points.

Theorem 1.2 (printed p. 254). "For every planar convex body $K$, $N(K)<\infty$."

The paper states that it has not found a finite upper bound valid for all $K$ (p. 254).

Theorem 2.1 (printed p. 255), the stronger version proved in § 2. A line $l$ is a normal of $K$ at $P\in\partial K$ when $P\in l$ and the line orthogonal to $l$ through $P$ supports $K$ at $P$, and $\Gamma=\{(Q,l):Q\in\partial K,\ l\text{ is a normal of }K\text{ at }Q\}$. "Given a convex body $K$, there is a point $P\in\partial K$ such that the number $M$ of pairs $(Q,l)\in\Gamma$ with $P\ne Q$ and $P\in l$ is finite."

Theorem 2.1 gives Theorem 1.2 through the observation on p. 255: if exactly $M$ pairs $(Q,l)\in\Gamma$ have $P\in l$ and $P\ne Q$, then every circle centred at $P$ meets $\partial K$ in at most $M+1$ points, so $N(K)\le M+1$. The paper adds (p. 255) that the proof shows $M$ finite on a part of the boundary of positive perimeter.

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; Theorem 1.2 on printed p. 254, Theorem 2.1 and the reduction on p. 255. The edition read is identified on the source card.

Read depth. Claims checked: both theorems, the definitions and the reduction were read clause by clause on the page images. The proof of Theorem 2.1 (pp. 255--257) was read for structure only, and nothing here is independently reviewed.

Proof pointer

§ 2, pp. 255--257. On the pairs $(Q,l)$ whose normal line meets $\partial K$ in exactly one further point $f(Q,l)$, the map $f$ is locally Lipschitz where the angle between $l$ and the boundary at $f(Q,l)$ exceeds $t$ (Lemma 2.2, p. 256). If $K$ is not a polygon with at most 6 sides, Lemma 2.3 (p. 257) gives a boundary set $F$ of positive perimeter whose preimage stays in such a region, and the coarea formula then shows that some $P\in F$ has finitely many preimages; for a polygon with at most 6 sides, $M\le12$ at every boundary point (p. 257). Not checked here.

Dependencies

Lemmas 2.2 and 2.3 of the paper; the coarea formula, cited to Federer, Geometric Measure Theory (1969).

Bears on

• Problem 982: the theorem bounds, for each convex body, the number $N(K)$ attached to Erdős's 1946 convex-curve statement, the strongest of the conjectures his paper poses on p. 248; it gives no bound uniform in $K$, concerns points of a convex curve rather than vertices of a polygon, and leaves the problem's statement undecided.