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Statement

$f(z)=\prod_{\nu=1}^n(z-z_\nu)$ and $E=\{|f(z)|\le1\}$ (p. 97). Pólya's theorem, recalled on p. 102, says the projection of a compact set of capacity 1 onto a line has linear measure $d\le4$. Quoted (p. 103): "The inequality $d\le4$ implies that the maximum of the measures of the different projections of $E$ is at most 4. Let $b$ be the minimum of the measures of the projections of $E$. By applying the approximation theorem to the '5-Stern' [10, p. 73] we obtain a lemniscate domain with $b>2.386$ (compare [2, Problem 10a]). I shall give an upper bound for $b$."

Theorem 7 (p. 103). "Let $F$ be a closed bounded set with $\operatorname{cap}F=1$. Then the projection of $F$ onto a certain straight line has measure less than $3.30$."

So there is a monic polynomial whose lemniscate set projects onto every line to a set of measure above $2.386$, while for every lemniscate set, which has capacity 1, some projection has measure below $3.30$. Problem 10a of the 1958 paper asks for a line onto which $\overline E$ projects to measure at most 2; the example answers it in the negative. The paper prints no details of the 5-Stern (a five-armed star of capacity 1 from the author's Math. Ann. 139 paper, [10, p. 73], not held) or of the value $2.386$.

Source. Ch. Pommerenke, On metric properties of complex polynomials, Michigan Math. J. 8 (1961), no. 2, 97--115; the passage and Theorem 7 with its proof on printed pp. 103--104 (PDF pp. 7--8 of the publisher's scan), Pólya's bound and Theorem 6 on p. 102 (PDF p. 6), read on the page images (the scan has no text layer). The copy read is identified in the source digest.

Read depth. Claims checked: the passage, Theorem 7 and Pólya's bound were read clause by clause on the page images; the proof of Theorem 7 (half a page) was read in full and followed, with its two external inputs taken as cited. The 5-Stern construction is not in the paper and was not read. Nothing here is independently reviewed.

Proof pointer

The example (p. 103) is a one-sentence application of the approximation theorem of p. 97 (a closed bounded set of capacity 1 lies in the interior of a lemniscate curve $\{|f|=\rho^n\}$, with $f$ monic and $\rho$ slightly above 1, and the curve lies in an $\varepsilon$-neighborhood of the set) to the 5-Stern, a set of capacity 1 whose minimal projection exceeds $2.386$; the projections of a close approximant exceed the same bound.

Theorem 7 (pp. 103--104): $F$ can be enclosed by closed curves $L_\mu$, $\mu=1,\ldots,m$, of total length $\sum\Lambda_\mu<10.36$ (the author's [12, Theorem 2]), taken convex by passing to convex hulls and merging those that intersect. With $b_\mu(\theta)$ the width of $L_\mu$ in direction $\theta$, $\Lambda_\mu=\frac12\int_0^{2\pi}b_\mu(\theta)\,d\theta$ (Bonnesen and Fenchel [1, p. 48]), so $\frac12\int_0^{2\pi}\sum_\mu b_\mu(\theta)\,d\theta<10.36$ and some $\theta_0$ has $\sum_\mu b_\mu(\theta_0)<10.36/\pi<3.30$, and $F$ projects onto the line of direction $\theta_0+\pi/2$ in a set of measure at most that sum.

Dependencies

The approximation theorem (p. 97, from the paper's [5]); the 5-Stern and its minimal projection from the author's Über die Kapazität ebener Kontinuen, Math. Ann. 139 (1959/60), 64--75, p. 73 (the paper's [10], not held); for Theorem 7, the perimeter bound $10.36$ for curves enclosing a capacity-1 set from the author's Einige Sätze über die Kapazität ebener Mengen, Math. Ann. 141 (1960), 143--152, Theorem 2 (the paper's [12], not held), and the Cauchy width formula from Bonnesen and Fenchel (1934).

Bears on

• Problem 1043: the negative answer. The problem asks for a line onto which $\{|f|\le1\}$ projects to measure at most 2; the example has every projection above $2.386$. Theorem 7 shows the largest possible minimal projection is below $3.30$, and Theorem 6 (p. 102) shows every projection of a degree-$n$ lemniscate set has measure at most $4\cdot2^{-1/n}$. The disproof rests on the unheld 5-Stern of the author's Math. Ann. 139 paper, not on the 1959 note filed as pommerenke_1959_some_problems_erdos_herzog_piranian, which this paper cites for Problem 10b (Theorem 2, p. 98); the site's status "DISPROVED (LEAN)" was not traced to a formal proof here.
• Problem 509: context only. A cover of $E$ by disks with radii summing to at most 2 projects onto every line to measure at most 4, which the example does not contradict; the paper states no result on the covering question.