Projections of polynomial hulls

The following theorem is discussed. Let X be a compact subset of the unit sphere in $\text{C}$ⁿ whose polynomially convex hull, $\text{X}\text{?}$, contains the origin, then the sum of the areas of the n coordinate projections of $\text{X}\text{?}$ is bounded below by π. This applies, in particular, when $\text{X}\text{?}$ 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 π; 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.