Barany 2013 question famous paper erdos

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theorem_1_1: Bárány and Roldán-Pensado construct a convex 15-gon K for which every boundary point is the centre of a circle meeting the boundary in at least 6 points while some boundary point has no circle meeting it in 7 or more points, so N(K) = 6, far above the value 2 of Erdős's 1946 convex-curve statement.

theorem_1_2: Bárány and Roldán-Pensado prove that every planar convex body has a boundary point P such that every circle centred at P meets the boundary in a bounded number of points, deduced from Theorem 2.1, which finds a boundary point lying on only finitely many normals of the body at other boundary points.

theorem_1_3: For every ε > 0 Bárány and Roldán-Pensado construct a convex body on whose boundary the points that are centres of circles meeting the boundary in infinitely many points make up more than a (1 − ε)-fraction of the perimeter.

theorem_1_4: In the Baire category sense, for most planar convex bodies K most points of the boundary are, for every n, centres of circles meeting the boundary in at least n points; the paper proves the stronger Theorem 4.1, with transversal intersections.

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Bárány, Imre and Roldán-Pensado, Edgardo, A question from a famous paper of Erdős. Discrete Comput. Geom. 50 (2013), no. 1, 253--261, doi:10.1007/s00454-013-9507-z. The copy read for this card is the publisher's PDF, which prints "© Springer Science+Business Media New York 2013", every other right reserved.

Source: https://link.springer.com/article/10.1007/s00454-013-9507-z. The article runs to 9 pages, printed pp. 253--261; received 20 September 2012, revised 9 April 2013, accepted 22 April 2013, published online 8 May 2013 (p. 253).

Read status: claims checked for the abstract and the introduction with the definitions of $N(K)$ and $J(K,n)$ and Theorems 1.1--1.4 (pp. 253--254), Theorem 2.1 and its reduction to Theorem 1.2 (p. 255), and the definitions of $J_0(K,n)$ and "most" with Theorem 4.1 (p. 260), each read clause by clause on the page images. The proofs (pp. 255--261) were read for structure only; none was checked, and nothing here is independently reviewed.

Contents

• § 1, Introduction (pp. 253--254). The paper quotes Erdős's 1946 statement that on every convex curve some point $P$ has every circle centred at $P$ meeting the curve in at most 2 points, notes that it fails for acute triangles and regular $(2k+1)$-gons, recalls that Erdős's related conjecture on a vertex with no three vertices equidistant from it was disproved by Danzer and by Fishburn and Reeds, defines $N(K)$ and $J(K,n)$, conjectures a bound on $N(K)$ independent of $K$, "probably by 6", and states Theorems 1.1--1.4.
• § 2, The finiteness of $N$ (pp. 255--257): normals and the curve $\Gamma$, the reduction of Theorem 1.2 to Theorem 2.1, Lemmas 2.2 and 2.3, and the proof of Theorem 2.1 by the coarea formula.
• § 3, Examples (pp. 257--260): Lemmas 3.1 and 3.2 (p. 258), the 15-gon of Theorem 1.1 (p. 259) and the bodies of Theorem 1.3 (pp. 259--260).
• § 4, Generic behaviour (pp. 260--261): transversal intersections, Theorem 4.1 and Lemma 4.2, proving Theorem 1.4.
• References (p. 261), six items, the first Erdős's 1946 paper (erdos_1946_sets_distances_points).

Bears on. #982: the paper studies the third of the conjectures Erdős poses on p. 248 of his 1946 paper, the convex-curve statement, which that paper states as stronger than the equidistance-free vertex conjecture from which it draws the problem's vertex bound. Theorem 1.1 shows that the bound 2 in the convex-curve statement must be raised to at least 6, and Theorem 1.2 that $N(K)$ is finite for each body. The results concern points of a convex curve and say nothing about distinct distances from a vertex of a convex polygon; the problem's statement is left undecided.

Results.

• Theorem 1.1 (p. 254): a planar convex body $K$, a 15-gon, with $N(K)=6$.
• Theorem 1.2 (p. 254): $N(K)<\infty$ for every planar convex body $K$, with the stronger Theorem 2.1 (p. 255) on its page.
• Theorem 1.3 (p. 254): for every $\varepsilon>0$ a convex body $K_\varepsilon$ with $|J(K_\varepsilon,\infty)|/|\partial K_\varepsilon|>1-\varepsilon$.
• Theorem 1.4 (p. 254): for most convex bodies, $\bigcap_nJ(K,n)$ contains most points of $\partial K$, with the stronger Theorem 4.1 (p. 260) on its page.

No file of this source is held: no license on record permits its redistribution, and the card cites the edition it names above.