Geodesic
On a~metric property of analytic sets
Izvestiya. Mathematics , Tome 10 (1976) no. 6, pp. 1333-1338.

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Let $H$ be an algebraic set in $\mathbf C^n$ containing the origin and let $S=\{z\in\mathbf C^n:|z|=1\}$ be the unit sphere. Conjecture. The diameter of one of the connected components of $H\cap S$ is greater than one. In this article it is shown that this is false if the requirement that $H$ be algebraic is weakened to the demand that the projections onto the coordinate planes be open. If, however, $S$ is replaced by the boundary of the unit polydisc, then the conjecture holds and the proof uses only the openness of the projection. Bibliography: 3 titles. 
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V. K. Beloshapka. On a~metric property of analytic sets. Izvestiya. Mathematics , Tome 10 (1976) no. 6, pp. 1333-1338. http://geodesic.mathdoc.fr/item/IM2_1976_10_6_a7/

[1] Ronkin L. I., Vvedenie v teoriyu tselykh funktsii mnogikh peremennykh, «Nauka», M., 1971 | MR | Zbl

[2] Lelong P., “Proprietés métriques des variétés analytiques complexes définies par une équation”, Ann. Scient. Ecole Supér, 67 (1950), 393–419 | MR | Zbl

[3] Springer Dzh., Vvedenie v teoriyu rimanovykh poverkhnostei, IL, M., 1960