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Statement

Section 3 of the survey (printed pp. 349–350) reviews conjectures of the earlier paper of Erdős, Herzog and Piranian, cited there as (I). On p. 350 Erdős takes complex numbers $z_1,\ldots,z_n$ with

$$|z_i-z_j|\le2\qquad(1\le i<j\le n)$$

and writes "We conjectured" (p. 350) that

$$\prod_{1\le i<j\le n}|z_i-z_j|$$

attains its maximum when the $z_i$ are the vertices of a regular polygon, which the print calls a regular polygon "of diameter 1" (p. 350). The passage itself cites no paper for the conjecture; the section opens with conjectures "in (I)", and the "we" are read here as the authors of (I). He reports that Danzer and Pommerenke (his reference [3], 1967) disproved the conjecture for even $n$, writes that it "probably holds for odd n" (p. 350), and adds that as far as he knows it is open for $n\ge5$.

The passage is a report of a conjecture and of its standing in 1976. The survey gives no proof, no counterconfiguration and no product computation for any of these assertions.

Source. P. Erdős, Extremal problems on polynomials, in Approximation Theory II (Academic Press, 1976), 347–355; Section 3, the unnumbered conjecture on printed p. 350 (PDF p. 4). The edition is identified in the source digest.

Read depth. Claims checked: the passage was read clause by clause on the page image. A conjecture has no proof to check; the normalization note below is the corpus's own.

Normalization

The constraint $|z_i-z_j|\le2$ and the words "of diameter 1" do not agree as printed. The product is positive and homogeneous of degree $n(n-1)/2$ in the configuration, so a regular polygon of diameter $1$ is beaten by the same polygon scaled to diameter $2$, which still satisfies the constraint. The scale-consistent reading compares with a regular $n$-gon of diameter $2$, and this is the reading Problem 1045 adopts.

Problem 1045 states the objective as $\prod_{i\ne j}|z_i-z_j|$, the square of the product displayed here; the two have the same maximizing configurations.

Dependencies

None. The passage cites Danzer and Pommerenke (1967) for the even-order disproof; the section's (I) is Erdős, Herzog and Piranian (1958), the survey's reference [7].

Bears on

• Problem 1045: this passage poses the problem's regular-polygon question, in the unordered-product normalization and with the diameter wording noted above, and reports the even-order disproof and Erdős's odd-order expectation as of 1976. It records no result of its own on the problem.