polynomially convex hull

Definition.

Let $K\subset ℂ$ be a compact subset of the complex plane,then the polynomially convex hull of $K$, denoted $\widehat{K}$, is defined as

$$\widehat{K}:=\{z\in ℂ:|p(z)|\le \underset{\zeta \in K}{\max }|p(\zeta )|\text{ for all polynomials }p\}.$$

A compact set $K$ is said to be polynomially convex if$K=\widehat{K}$

Obviously $K\subset \widehat{K}$. The intuitive idea behind this definition is that the polynomially convex hull of $K$ fills in any “holes” that may exist in $K$.The following proposition makes that precise.

Proposition.

If $K\mathrm{\subset }\mathrm{C}$ is a polynomially convex set, thenall the components of the interior of $K$ are simply connected.

One of the reasons for this definition is the following result.

Proposition.

Let $f$ be a function analytic in an open neighbourhood $N$ of a compact set$K\mathrm{\subset }\mathrm{C}$, and suppose that $f$ can be approximated by polynomials uniformlyon compact subsets of $N$. Then $f$ can be extended analytically to a neighbourhood of $\widehat{K}$.

For example if we take $K=\{z\in ℂ:|z|=1\}$(the unit circle)then $\widehat{K}=\{z\in ℂ:|z|\le 1\}$ (theclosed unit disc). The fact that the inside of the disc belongs to$\widehat{K}$ follows from the maximum modulus principle as polynomials are analytic functions. The fact that $\widehat{K}$ does not contain anything outside theclosed unit disc follows by looking at the polynomial $p(z)=z$ which hasalways greater modulus outside of the unit disc then anywhere on the unit circle. So if we have a function defined on a neighbourhood of the unit circleand which we can approximate uniformly on compact subsets of thisneighbourhoodby polynomials, then we can extend this function analyticallyto the whole unit disc. So this for example implies that $f(z):=\frac{1}{z}$cannot be approximated uniformly on compact subsets by polynomials on aneighbourhood of the unit circle.

The reason why we call $\widehat{K}$ a “hull” of some is that the conventional convex hull of $K\subset {ℝ}^n$ can be defined as the set of points $x$ such that for all linear functions $f:{ℝ}^n\to ℝ$ we have $|f(x)|\le {sup}_{y\in K}|f(y)|$. This coincides with conventional definition because if $x$ is not in the conventional convex hull, then there is a linear functional that separates $x$ from the hull (by Hahn-Banach theorem in general, or more elementarily in ${ℝ}^n$ by Farkas’s lemma), and conversely if $x$ is in the convex hull of $K$ then such linear function does not exist for the same reason. So, intuitively the conventional convex hull is set of point that are inseparable from from $K$ by linear functions. Polynomially convex hull is the same thing, but with polynomials.Of course similar definitions can be made with respect to other classes of functions. For example, hulls with respect to plurisubharmonic functions are very useful in multivariate complex analysis.