---

Statement

Notation (printed pp. 254 and 260): $\mathcal K$ is the set of planar convex bodies with the Hausdorff metric; $J(K,n)$ is the set of points $P\in\partial K$ for which some circle centred at $P$ meets $\partial K$ in at least $n$ points. In a Baire space, "most points" satisfy a property when the set of points satisfying it contains a dense $G_\delta$ set (p. 260).

Theorem 1.4 (printed p. 254). "For most convex bodies $K\in\mathcal K$, the set

$$\bigcap_{n\in\mathbb N}J(K,n)$$

contains most points of $\partial K$."

Theorem 4.1 (printed p. 260), the stronger statement proved. A circle $\mathcal S$ meets $\partial K$ transversally at $Q$ when every neighbourhood of $Q$ contains points of $\mathcal S$ in the interior of $K$ and points of $\mathcal S$ outside $K$; $J_0(K,n)\subset J(K,n)$ is the set of $P\in\partial K$ for which some circle centred at $P$ meets $\partial K$ transversally in at least $n$ points. "For most convex bodies $K\in\mathcal K$, the set $\bigcap_{n\in\mathbb N}J_0(K,n)$ contains most points of $\partial K$."

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 1.4 on printed p. 254, the definitions and Theorem 4.1 on p. 260. The edition read is identified on the source card. The acknowledgments (p. 261) credit Rolf Schneider with suggesting the theorem and with a different proof.

Read depth. Claims checked: both theorems and the definitions were read clause by clause on the page images. The proof (pp. 260--261) was read for structure only, and nothing here is independently reviewed.

Proof pointer

§ 4, pp. 260--261. Let $\mathcal K_{n,m}$ be the bodies $K$ such that every $P\in\partial K$ has a point of $J_0(K,n)$ within distance $\frac1m$. Lemma 4.2 (p. 260) shows each $\mathcal K_{n,m}$ open and dense in $\mathcal K$, density by replacing far vertices of an approximating polygon with $n$ vertices on a circle about a side's midpoint. The intersection over $n,m$ is then a dense $G_\delta$, and for $K$ in it each $J_0(K,n)$ is open and dense in $\partial K$. Not checked here.

Dependencies

Lemma 4.2 of the paper; the Baire category theorem, cited to Schechter, Handbook of Analysis and Its Foundations (1997), Chap. 20.

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.