Quasi-isoperimetric inequality for polynomial hulls (Q1315259)

If \(X\) is a smooth curve in \(\mathbb{C}^ n\) which bounds a relatively compact, complex analytic, 1-dimensional subvariety \(V\) of \(\mathbb{C}^ n \backslash X\), then the isoperimetric inequality is the statement that \(4 \pi \text{area} (V) \leq [\text{length} (X)]^ 2\). For a general compact set \(X\) in \(\mathbb{C}^ n\), it would be replaced with \(4 \pi {\mathcal H}^ 2 (\widehat X) \leq [{\mathcal H}^ 1 (X)]^ 2\), where \({\mathcal H}^ k\) denotes the \(k\)-dimensional Hausdorff measure and \(\widehat X\) is the polynomial convex hull of \(X\). The author showed that such an inequality holds if \(X\) is a Jordan curve [Am. J. Math. 110, No. 4, 629-640 (1988; Zbl 0659.32017)]. However, it is not true in general [same author, J. Anal. Math. 47, 238-242 (1986; Zbl 0615.32009)]. In this paper a weaker inequality is established, which is valid for any compact set \(X\) in \(\mathbb{C}^ n\): For almost all unitary coordinate systems \(p_ 1\), \(p_ 2, \dots, p_ n\) of \(\mathbb{C}^ n\), one has \(4 \pi \sum^ n_{k=1} {\mathcal H}^ 2 (p_ k (\widehat X)) \leq [{\mathcal H}^ 1 (X)]^ 2\). This yields the inequality \({\mathcal H}^ 1(X) \geq 2 \pi \text{dist} (a,X)\) for any \(a \in \widehat X\). In particular, if \(X\) lies in the unit sphere and if \(0 \in \widehat X\), then \({\mathcal H}^ 1(X) \geq 2 \pi\), which gives a solution to a problem raised by \textit{E. L. Stout} [Problem 8.4.2, Several Complex Variables Problem List, J. E. Fornaess et al., eds., Ann Arbor, 1991] and solved earlier by Forstnerič in the case where \(X\) is \(({\mathcal H}^ 1,1)\) rectifiable. The author also proves that if \(X\) is a compact, totally disconnected subset of \(\mathbb{C}^ n\) and if (*) \(X\) is a purely \(({\mathcal H}^ 1,1)\) unrectifiable set, then \(X\) is polynomially convex. It is an open question whether this remains true when the hypothesis (*) is dropped. If, however, (*) is replaced by the assumption that \(X\) is 1 rectifiable, then the conclusion remains valid.