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Statement

Context (Section 8, p. 175). Erdős recalls his conjecture, proved by Altman (the paper's [7]), that the vertices of a convex $n$-gon determine at least $[n/2]$ distinct distances, and his further conjecture, still open as far as he knows, that some vertex of a convex $n$-gon has at least $[n/2]$ distinct distances to the others.

The refuted conjecture (p. 175). Erdős had also conjectured that every convex $n$-gon has a vertex with no three other vertices equidistant from it. Danzer disproved it; his example (Fig. 5, p. 175) is a convex nonagon $A_1B_1C_1A_2B_2C_2A_3B_3C_3$ with threefold rotational symmetry and

$$A_1A_2=A_1A_3=A_1B_3,\qquad B_1B_2=B_1C_2=B_1B_3,\qquad C_1C_2=C_1A_3=C_1C_3,$$

so that by the symmetry every vertex has three other vertices at a common distance from it.

The construction (pp. 175-176), in outline. Start from a Reuleaux triangle $A_1A_2A_3$, extend the arc $A_3A_1$ beyond $A_1$ to a point $B_1$ close to $A_1$, define $B_2,B_3$ by the symmetry, and draw the Reuleaux triangle $B_1B_2B_3$. With $B_i'$ the midpoint of the side $B_iB_{i+1}$ ($B_4=B_1$), choose $C_1$ on the arc $B_1B_1'$ and $C_2,C_3$ by the symmetry. At $C_1=B_1$ one has $C_1C_3>C_1A_3$, and at $C_1=B_1'$ one has $C_1C_3<C_1A_3$ provided $B_1A_1$ is sufficiently small, so an intermediate position gives $C_1C_2=C_1C_3=C_1A_3$.

Question (p. 176). "Perhaps in every convex polygon there is a vertex which does not have four other vertices equidistant from it." Erdős poses it without a proof or a counterexample.

Szemerédi's conjecture (p. 176). The section ends with Szemerédi's conjecture that $n$ points with no three on a line determine at least $[n/2]$ distinct distances, which Szemerédi can prove only with $[n/3]$.

Source. P. Erdős, Some combinatorial and metric problems in geometry, Intuitive geometry (Siófok, 1985), Colloq. Math. Soc. János Bolyai 48, North-Holland, Amsterdam-New York, 1987, 167--177 (MR 89i:52012); Section 8, printed pp. 175-176, with Fig. 5 on p. 175.

Read depth. Claims checked: the conjectures, the distance relations of the nonagon, the construction and the four-vertex question were read clause by clause on the page images of pp. 175-176. The construction's convexity and the intermediate-value step were not re-derived here; the paper prints no coordinates.

Proof pointer

Pages 175-176: the construction outlined above, an intermediate-value argument in the position of $C_1$ on the arc $B_1B_1'$. A nine-point convex set realizing the three relations, with exact evidence, is recorded on the nonagon page; it is not identified as Danzer's own choice.

Dependencies

Altman, Canad. Math. Bull. 15 (1972), 329--340 (the paper's [7]), for the $[n/2]$ bound recalled as context.

Bears on

• Problem 97: the question is the site's statement, whether every convex polygon has a vertex with no other four vertices equidistant from it; Danzer's nonagon answers the earlier three-vertex form in the negative and leaves the four-vertex question open in this paper.