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Claim. Let $\rho_n=\inf\{\rho(\Lambda_p):p\in\mathcal P_n(\mathbb D)\}$ be the minimal inradius of the lemniscate $\Lambda_p=\{z:\lvert p(z)\rvert<1\}$ over monic polynomials $p$ of degree $n$ with all zeros in the closed unit disk, the quantity $\inf_{\deg f=n}\rho(f)$ of Problem 1039. Theorem 8 of Krishnapur, Lundberg and Ramachandran, On the area of polynomial lemniscates, arXiv:2503.18270 (2025-03-24), states that

$$\rho_n\ge\frac{c}{n\sqrt{\log n}}$$

for an absolute constant $c>0$. The paper derives it from its Theorem 1, the lower bound of order $1/\log n$ for the minimal area of $\Lambda_p$, through its Lemma 9, a quantitative form of the Cuenya–Levis inequality $\rho(\Lambda_p)\ge(c'/n)\sqrt{m(\Lambda_p)}$, which also confirms the conjecture of Solynin and Williams that the constant there is of order $1/n$. The authors present the bound as improving Pommerenke's $1/(2en^2)$ and as supporting, up to the logarithm, the $1/n$ rate that Erdős, Herzog and Piranian asked about. The paper is digested on the card Krishnapur, Lundberg and Ramachandran 2025.

Covers. The lower bound $\rho_n\gg1/(n\sqrt{\log n})$, which improves Pommerenke 1961. It settles neither the order of $\rho_n$ nor the second question, whether $\rho_n\gg1/n$; the order $1/n$ is the later claim on Price 2026.

Depends on. No page of this wiki.

Standing. Claimed. The preprint is at its first arXiv version and no journal version is recorded, so no refereed is listed. The site's commentary credits the bound to the paper, cited as [KLR25], but labels the problem OPEN, so the credit is not acceptance and no reviewed is listed. The proof is not verified by this corpus.