Let $K,K_{0}$ be as in the theorem. Observe that $K_{0}$ is a circled subset of $\partial\mathbb{B}^{n+1}$: if $z\in K_{0}$ then $e^{i\theta}z\in K_{0}$, $\forall\theta\in[0,2\pi]$. For such compacts, the polynomial hull $\widehat{K_{0}}$ coincides with the ”homogeneous polynomial hull”,

$$\widehat{K_{0}}^{h}:=\{x\in\mathbb{C}^{n+1}\,/\,|P(x)|\leq\sup_{F}|P|,\,\forall P\text{ homogeneous polynomial}\}.$$

Indeed one inclusion $\widehat{K_{0}}\subset\widehat{K_{0}}^{h}$ is clear, so assume $z_{0}\in\widehat{K_{0}}^{h}$. Let $P=\sum_{j=0}^{d}P_{j}$ be a polynomial of degree $d$ decomposed into its homogenous components. Observe that $P_{j}(x)=(2\pi)^{-1}\int_{0}^{2\pi}P(e^{i\theta}x)e^{-ij\theta}d\theta$. Therefore $\sup_{K_{0}}|P_{j}|\leq\sup_{K_{0}}|P|$ since $K_{0}$ is circled. Fix $t\in]0,1[$. Then

$$|P(tz_{0})|\leq\sum_{j=0}^{d}t^{j}|P_{j}(z_{0})|\leq\frac{1}{1-t}\sup_{K_{0}}|P|.$$

We infer $tz_{0}\in\widehat{K_{0}}$. Letting $t\rightarrow 1^{-}$ and using that $K_{0}$ is closed we get $z_{0}\in\widehat{K_{0}}$, whence $\widehat{K_{0}}=\widehat{K_{0}}^{h}$.

Fix now $z\in\mathbb{C}^{n+1}$ such that $||z||\leq T_{\omega}(K)$. Let $P$ be a homogeneous polynomial of degree $d$. Then

(2)

$$|P(z)|=||z||^{d}\left|P\left(\frac{z}{||z||}\right)\right|\leq T_{\omega}(K)^{d}\sup_{\partial\mathbb{B}^{n+1}}|P|$$

Now set $\psi(z)=d^{-1}\log|P(z)|-\log||z||$ and $\varphi=\psi-\sup_{K}\psi$. Then $\varphi\in PSH(X,\omega)$ with $\sup_{K}\varphi\leq 0$ hence $\varphi\leq V_{K,\omega}$. Therefore

$$T_{\omega}(K)^{d}\leq\exp(-d\sup_{\mathbb{C}\mathbb{P}^{n}}\varphi)=\frac{\sup_{K_{0}}|P|}{\sup_{\partial\mathbb{B}^{n+1}}|P|}.$$

Together with $(2)$ this yields $|P(z)|\leq\sup_{K_{0}}|P|$ hence $z\in\widehat{K_{0}}^{h}=\widehat{K_{0}}$. Thus $K_{0}$ contains the ball centered at the origin of radius $T_{\omega}(K)$.

Conversely since $T_{\omega}(K)=T_{\omega}^{\prime}(K)$ (theorem 4.1), one can find homogenous polynomials $P_{j}$ of degree $d_{j}$ such that $\sup_{\partial\mathbb{B}^{n+1}}|P_{j}|^{-1/d_{j}}\cdot\sup_{K_{0}}|P_{j}|^{1/d_{j}}\rightarrow T_{\omega}(K).$ Assume $r\mathbb{B}^{n+1}\subset\widehat{K_{0}}$. Then

$$r^{d_{j}}\sup_{\partial\mathbb{B}^{n+1}}|P_{j}|=\sup_{r\mathbb{B}^{n+1}}|P_{j}|\leq\sup_{K_{0}}|P_{j}|$$

yields $r\leq T_{\omega}(K)$. ∎