A New Angle on L2 Regularization

Imagine two high-dimensional clusters and a hyperplane separating them. Consider in particular the angle between: the direction joining the two clusters' centroids and the normal to the hyperplane. In linear classification, this angle depends on the level of L2 regularization used. Can you explain why?
Comment: An explorable explanation on the phenomenon of adversarial examples in linear classification and its relation to L2 regularization. Interactive version available at: https://thomas-tanay.github.io/post--L2-regularization/