Further regularity results in \(P-\)extremal setting

In our paper, Pluripotential theory and convex bodies: a Siciak-Zaharjuta theorem (Computational Methods and Function Theory (2020), 20(3-4), 571-590) by Bayraktar et al., we proved a regularity result of \(P-\) $P-$ extremal function. In this paper, we prove some other stronger regularity results analogue to those in the standard setting. Specifically we focus on proving the following proposition: Proposition: Let \(K={\bar{D}}\subset {\mathbb {C}}^d\) $K=\overline{D}\subset {\mathbb{C}}^d$ be the closure of a bounded domain D with \(\partial D\) $\partial D$ of class \(C^{1,1}\) $C^{1,1}$ . Then, for any \(Q\in {\mathcal {C}}_\alpha (K)\) $Q\in {\mathcal{C}}_{\alpha }(K)$ , where \({\mathcal {C}}_\alpha (K)\) ${\mathcal{C}}_{\alpha }(K)$ is the Hölder class \(\alpha\) $\alpha$ on K, we have \(V_{P,K,Q}\in {\mathcal {C}}_\alpha ({\mathbb {C}}^d) \text { for a convex body } P\subset (\mathbb {R^+})^d.\) $V_{P,K,Q}\in {\mathcal{C}}_{\alpha }({\mathbb{C}}^d)\text{for a convex body}P\subset {({\mathbb{R}}^{+})}^d.$