Multidimensional cost geometry

Abstract

In this paper, we study the geometric structure induced by the canonical reciprocal cost function and its natural $n$-dimensional extension. In logarithmic coordinates, the potential depends only on the linear combination $S=\alpha\cdot t$, and the associated Hessian metric has rank one at every point. The geometry is intrinsically degenerate and effectively one-dimensional, with an $(n-1)$-dimensional null distribution. On the other hand, when the same function is expressed in the original $x$-coordinates, the corresponding Hessian is generically nondegenerate and defines a pseudo-Riemannian metric away from explicit singular hypersurfaces. We further analyze affine and Levi-Civita geodesics and compare their behavior. In particular, affine geodesics in logarithmic coordinates are globally defined, while in $x$-coordinates their behavior is restricted by the domain and the singular set. Finally, we relate the construction to symmetrized Itakura-Saito and Bregman divergences, and give a Fisher-Rao realization of the logarithmic Hessian metric.