  Adversarial robustness is a critical property in a variety of modern machine
learning applications. While it has been the subject of several recent
theoretical studies, many important questions related to adversarial robustness
are still open. In this work, we study a fundamental question regarding Bayes
optimality for adversarial robustness. We provide general sufficient conditions
under which the existence of a Bayes optimal classifier can be guaranteed for
adversarial robustness. Our results can provide a useful tool for a subsequent
study of surrogate losses in adversarial robustness and their consistency
properties. This manuscript is the extended and corrected version of the paper
\emph{On the Existence of the Adversarial Bayes Classifier} published in
NeurIPS 2021. There were two errors in theorem statements in the original paper
-- one in the definition of pseudo-certifiable robustness and the other in the
measurability of $A^\e$ for arbitrary metric spaces. In this version we correct
the errors. Furthermore, the results of the original paper did not apply to
some non-strictly convex norms and here we extend our results to all possible
norms.
