Projections of polynomial hulls

Abstract

The following theorem is discussed. Let X be a compact subset of the unit sphere in n whose polynomially convex hull, , contains the origin, then the sum of the areas of the n coordinate projections of is bounded below by [pi]. This applies, in particular, when is a one-dimensional analytic subvariety V containing the origin, and in this case generalizes the fact that the "area" of V is at least [pi]; in fact, the area of V is the sum of the areas of the n coordinate projections when these areas are counted with multiplicity. A convex analog of the theorem is obtained. Hartog's theorem that separate analyticity implies analyticity, usually proved with the use of subharmonic functions (Hartog's lemma), will be derived as a consequence of the theorem, the proof of which is based upon the elements of uniform algebras.