Several Complex Variables Problem List — Problem 22

Definitions and mathematical statement

Let $n\geq 1$ be an integer. A set $X\subset{\mathbb{C}}^n$ is totally disconnected if every connected subset of $X$ has at most one point. With respect to the Euclidean metric on ${\mathbb{C}}^n$, its one-dimensional Hausdorff measure is \[ \mathcal H^1(X)=\lim_{\delta\downarrow0}\inf\left\{\sum_{j=1}^{\infty}\operatorname{diam}(U_j): X\subseteq\bigcup_{j=1}^{\infty}U_j,\ \operatorname{diam}(U_j)<\delta\text{ for all }j\right\}. \] For a nonempty compact set $X\subset{\mathbb{C}}^n$, define its polynomial hull by \[ \widehat X=\left\{z\in{\mathbb{C}}^n: |p(z)|\leq\max_{x\in X}|p(x)|\text{ for every }p\in{\mathbb{C}}[z_1,\ldots,z_n]\right\}. \] Set $\widehat\varnothing=\varnothing$. A compact set is polynomially convex if $\widehat X=X$.

Question (H. Alexander). For every integer $n\geq1$ and every compact, totally disconnected set $X\subset{\mathbb{C}}^n$ satisfying $\mathcal H^1(X)<\infty$, must $\widehat X=X$? The measure hypothesis permits $\mathcal H^1(X)=0$; no positivity assumption is imposed.

Short English statement

Must every compact, totally disconnected subset of ${\mathbb{C}}^n$ with finite one-dimensional Hausdorff measure be polynomially convex? The supplied registry labels this question open; that label is not an independently verified assessment.

Sources

• Local UnsolvedMath registry, permanent ID $800226$, source code catalogue entry, titled Several Complex Variables Problem List --- Problem 22. The supplied record lists Buzzard, Fornaess, and Krantz in its proposer field and gives the year 1995.
• Buzzard, Fornaess, and Krantz, The Several Complex Variables Problem List (1995), Problem 22. Supplied source URL: https://arxiv.org/abs/math/9509202. The source statement attributes the question to H. Alexander and supplies the contact U22330@UICVM.bitnet.
• Supplied provenance records access on 2026-10-16 and extraction by arxiv_latex. This access information is supplied provenance, not independent source verification performed for this draft. The supplied registry status is open.