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The length of a set in the sphere whose polynomial hull contains the origin. (English) Zbl 0762.32007
Definition. (a) A set $$X<\mathbb{R}^ k$$ is 1-rectifiable if it is the image of a bounded subset $$U\subset\mathbb{R}$$ under a Lipschitz continuous mapping $$f:U\to\mathbb{R}^ k$$.
(b) $$X$$ is $$({\mathcal H}^ 1,1)$$-rectifiable if $${\mathcal H}^ 1(X)<\infty$$ and $${\mathcal H}^ 1$$-almost all of $$X$$ can be covered by a countable union of 1-rectifiable sets.
$${\mathcal H}^ 1$$ denotes here a 1-dimensional Hausdorff measure.
The main result of the paper is Theorem. If $$X$$ is a compact $$({\mathcal H}^ 1,1)$$-rectifiable subset of the unit sphere $$S\subset\mathbb{C}^ n$$ such that the origin $$0\in\mathbb{C}^ n$$ belongs to the polynomial hull $$\widehat X$$, then $${\mathcal H}^ 1(X)\geq 2\pi$$.

##### MSC:
 3.2e+21 Polynomial convexity, rational convexity, meromorphic convexity in several complex variables
##### Keywords:
set in the sphere; polynomial hull
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##### References:
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