The present paper is devoted to a systematic study of the $p$-Brownian convergence introduced in \cite{boudabra2026stability} (in press) to study the stability of the planar Skorokhod embedding problem \cite{gross2019,Boudabra2020}. The first part is an illustration of some geometric aspects of the $p$-Brownian convergence. The second part turns this notion into a metric between domains. More precisely, we place it within the framework of optimal transport theory. Several results are obtained, namely asymptotic behavior in case of homothetic domains. Numerical illustrations are provided as well.
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