The proof of Theorem 3.14 contains an unsubstantiated claim. To overcome this problem, we add a hypothesis to the statement of 3.14 and we provide a new valid proof. We adjust Theorem 3.15, Corollary 3.16, Proposition 4.23, Theorem 4.26, Corollary 4.29, and Corollary 4.32 accordingly.

We provide a correspondence between one-sided coideal subrings and one-sided ideal two-sided coideals in an arbitrary bialgebroid. We prove that, under some expected additional conditions, this correspondence becomes bijective for Hopf algebroids. As an application, we investigate normal Hopf ideals in commutative Hopf algebroids (affine groupoid schemes) in connection with the study of normal affine subgroupoids.

We study a quasiconvex conjugation that transforms the level sum of functions into the pointwise sum of their conjugates and derivenew duality results for the minimization of the max of two quasiconvex functions. Following Barron and al., we show that the levelsum provides quasiconvex viscosity solutions for Hamilton-Jacobi equations in which the initial condition is a general continuousquasiconvex function which is not necessarily Lipschitz or bounded.