We prove that three pairwise disjoint, convex sets can be found, all congruent to a set of the form , such that their union has a non-trivial polynomial convex hull. This shows that not all holomorphic functions on the interior of the union can be approximated by polynomials in the open-closed topology.
@article{bwmeta1.element.bwnjournal-article-cmv70i1p7bwm,
author = {Backlund, Ulf and F\"allstr\"om, Anders},
title = {The polynomial hull of unions of convex sets in $$\mathbb{C}$^n$
},
journal = {Colloquium Mathematicae},
volume = {70},
year = {1996},
pages = {7-11},
zbl = {0844.32012},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-cmv70i1p7bwm}
}
Backlund, Ulf; Fällström, Anders. The polynomial hull of unions of convex sets in $ℂ^n$
. Colloquium Mathematicae, Tome 70 (1996) pp. 7-11. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-cmv70i1p7bwm/
[000] [BePi] E. Bedford and S. Pinchuk, Domains in with noncompact automorphism group, J. Geom. Anal. 1 (1991), 165-192.
[001] [Kal] E. Kallin, Polynomial convexity: The three spheres problem, in: Proceedings of the Conference on Complex Analysis, Minneapolis 1964, H. Röhrl, A. Aeppli and E. Calabi (eds.), Springer, 1965, 301-304.
[002] [Khud] G. Khudaĭberganov, On polynomial and rational convexity of unions of compacts in , Izv. Vuz. Mat. 2 (1987), 70-74 (in Russian).
[003] [KyKh] A. M. Kytmanov and G. Khudaĭberganov, An example of a nonpolynomially convex compact set consisting of three non-intersecting ellipsoids, Sibirsk. Mat. Zh. 25 (5) (1984), 196-198 (in Russian).
[004] [Ros] J.-P. Rosay, The polynomial hull of non-connected tube domains, and an example of E. Kallin, Bull. London Math. Soc. 21 (1989), 73-78. | Zbl 0634.32011
[005] [Wer1] J. Wermer, Polynomial approximation on an arc in , Ann. of Math. 62 (1955), 269-270.
[006] [Wer2] J. Wermer, An example concerning polynomial convexity, Math. Ann. 139 (1959), 147-150. | Zbl 0094.28302