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Statement

Notation (printed p. 254): for $n\in\mathbb N\cup\{\infty\}$, $J(K,n)$ is the set of points $P\in\partial K$ such that there is a circle centred at $P$ that meets $\partial K$ in at least $n$ points, and $|X|$ is the 1-dimensional Hausdorff measure (perimeter) of $X\subset\mathbb R^2$.

Theorem 1.3 (printed p. 254). "Let $\varepsilon>0$, then there is a convex body $K_\varepsilon$ such that

$$\frac{|J(K_\varepsilon,\infty)|}{|\partial K_\varepsilon|}>1-\varepsilon."$$

The paper adds (p. 254) that if $K_0$ is a segment or an acute triangle, $K_\varepsilon$ can be built so that $K_\varepsilon\to K_0$ in the Hausdorff metric as $\varepsilon\to0$. It suggests that part of the difficulty of finding a bound on $N(K)$ uniform in $K$ may come from this theorem and Theorem 1.4.

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 1.3 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 (pp. 259--260) was read for structure only, and nothing here is independently reviewed.

Proof pointer

§ 3, pp. 259--260. Near a triangle $A_1A_2A_3$, choose $B_i$ close to $A_i$ so that $A_1B_1A_2B_2A_3B_3$ is a convex hexagon with acute angles $\angle A_iB_iA_{i+1}$, and apply Lemma 3.1 (p. 258) with $N=\infty$ on each $A_iB_iA_{i+1}B_{i+1}$; near a segment $[A,B]$ the same is done with a convex quadrilateral $ACBD$ with acute angles at $C$ and $D$. Not checked here.

Dependencies

Lemma 3.1 (p. 258) of the paper.

Bears on

• Problem 982: the theorem concerns the circles centred at points of a convex curve in Erdős's 1946 convex-curve statement, the strongest of the conjectures his paper poses on p. 248; it says nothing about distinct distances from a vertex of a convex polygon and leaves the problem's statement undecided.