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In a recent paper [17] we studied asymmetric metric spaces; in this context we studied the length of paths, introduced the class of run-continuous paths; and noted that there are different definitions of “length spaces” (also known as “path-metric spaces” or “intrinsic spaces”). In this paper we continue the analysis of asymmetric metric spaces.We propose possible definitions of completeness and (local) compactness.We define the geodesics using as admissible paths the class of run-continuous paths.We...

We formulate an Hamilton-Jacobi partial differential equation $$H(x,Du(x\left)\right)=0$$ on a $n$ dimensional manifold $M$, with assumptions of convexity of $H(x,·)$ and regularity of $H$ (locally in a neighborhood of $\{H=0\}$ in $T^{*}M$); we define the “min solution” $u$, a generalized solution; to this end, we view $T^{*}M$ as a symplectic manifold. The definition of “min solution” is suited to proving regularity results about $u$; in particular, we prove in the first part that the closure of the set where $u$ is not regular may be...

This errata corrects one error in the 2004 version of this paper [Mennucci, (2004) 426–451].