Geodesic
A~criterion for the uniqueness of supports of analytic functionals in the class of $n$-circular convex compact sets
Izvestiya. Mathematics , Tome 63 (1999) no. 2, pp. 331-366.

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In this paper we give necessary and sufficient conditions for an $n$-circular convex compact set $K\subset\mathbb C^n$ to have the following uniqueness property: $K$ is the unique convex support for every analytic functional which has $K$ as a support. 
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A. S. Krivosheev. A~criterion for the uniqueness of supports of analytic functionals in the class of $n$-circular convex compact sets. Izvestiya. Mathematics , Tome 63 (1999) no. 2, pp. 331-366. http://geodesic.mathdoc.fr/item/IM2_1999_63_2_a3/

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