Polynomial Convexity: The Three Spheres Problem

Abstract

A compact subset Y of C ⁿ is called polynomially convex if for each point x in C ⁿ ~ Y there is a polynomial p such that

$$ \left| {p(x)} \right| > \sup \,\left\{ {\left| {p(y)} \right|:y \in Y} \right\} $$

we say that such a polynomial p separates x from Y. A set Y is convex in the ordinary sense just in case each point of the complement can be separated from Y by a polynomial of degree one, so each convex set is polynomially convex. Finite sets are also polynomially convex, so polynomially convex sets need not be connected. Any compact set which lies entirely in the real subspace of points having all real coordinates is polynomially convex because of the Weierstrass approximation theorem. Polynomial convexity, unlike ordinary convexity, is not preserved under real linear transformations or even under general complex linear mappings, though of course it is preserved by complex linear isomorphisms. There is no nice internal description of polynomially convex sets analogous to the condition that a set is convex if and only if it contains the closed line segment joining each pair of its points. Thus although many sets are known to be polynomially convex it can be rather difficult to decide about some fairly simple sets.

Received July 16, 1964.