Is the property of being polynomially convex a topological invariant?


In other words, let $M$ and $N$ be two homeomorphic, compact subsets of $n$-dimensional complex Euclidean space, and assume in addition that $M$ is polynomially convex. 



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> Is $N$ necessarily polynomial convex?
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Is polynomial convexity a topological invariant?
The answer is *no*. 


In fact, Kallin has shown in **[Kal64]** that the union of three disjoint closed balls is polynomially convex, but the union of three disjoint closed polydisks needs not to be polynomially convex.


Actually, it turns out that polynomial convexity is not even preserved by (real) linear transformations. For instance, Khudaĭberganov and Kytmanov have given in **[KK84]** an example of three closed disjoint ellipsoids the union of which is not polynomially convex.


**References**


**[Kal64]** E. Kallin: *[Polynomial convexity: The three spheres problem](http://link.springer.com/chapter/10.1007%2F978-3-642-48016-4_26#page-1)*, Proc. Conf. Complex Analysis (Minneapolis, 1964), Springer, Berlin, pp. 301–304, doi:10.1007/978-3-642-48016-4\_26.


**[KK84]** G. Khudaĭberganov and A. M. Kytmanov: *An example of a nonpolynomially convex compact set consisting of three non-intersecting ellipsoids*, Sibirsk. Mat. Zh. 25 (5) (1984), 196-198 (in Russian). 


