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Let $U\subset\mathbb{C}^n$ be a domain of holomorphy, then we know that $$d_{\infty}(K,\partial U)=d_{\infty}(\hat{K}_U,\partial U)$$

for each compact subset $K\subset U$, also that $$d_{2}(K,\partial U)=d_{2}(\hat{K}_U,\partial U)$$

for each compact subset $K\subset U$, this equality also holds for all norm on $\mathbb{C}^n$?

Any help would be appreciated.

$\hat{K}_U= \{z \in U: |f(z)| \leq \sup_K |f|, \forall f\in \mathcal{O}(\Omega)\}$: holomorphically convex hull of $K$.

$d_{\infty}(A,B)=\mathrm{inf}\{\Arrowvert a-b\Arrowvert_{\infty}:a\in A,b\in B\}$

$d_{2}(A,B)=\mathrm{inf}\{\Arrowvert a-b\Arrowvert_{2}:a\in A,b\in B\}$

$\Arrowvert z\Arrowvert_{2}=\left\{\sum_{j=1}^{n}|z_j|^2\right\}^{1/2}$ , $\Arrowvert z\Arrowvert_{\infty}=\max\{|z_1|,\ldots,|z_n|\}$: particular norms.

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felipeuni
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  • This is a good question, and one I was asking myself. I do not think it is true. If you look in Shabat's text, he proves this result with respect to the polydisk norm, which is the norm you provide above. What reference are you using? – AmorFati Nov 23 '17 at 07:13

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