Definition. A set E ⊂ Rn in real Euclidean space is called to be m-convex (m < n) if for each x ∈ Rn \ E there exists an m-dimensional plane L(x) such that x ∈ L(x) and L(x) ∩ E = ∅.
In a multidimensional complex space Cn, let us consider a set and its supplement such that a complex hyperplane drawn through every point of the supplement lies completely in this supplement. Such set is called linearly convex.
Problem. (problem of sphere). Does there exist a linearly convex compact set in the space C2 (or 2- convex compact set in R4) which is homeomorphic to a two-dimensional sphere S2?
An interesting problem, thank you!
Just one remark. You did not mention that the complex and real versions of the problem are not equivalent, because not every real linear subspace of real dimension 2 in C2 is at the same time a one-dimensional complex subspace. So in fact you pose not one problem but two different problems :)
You are right. But problems are connected. More more detail in my book
"Convexity..."
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