We re-examine measure-category duality by a bitopological approach, using both the Euclidean and the density toplologies of the line. We give a topological result (on convergence of homeomorphisms to the identity) obtaining as a corollary results on in…nitary combinatorics due to Kestelman and to Borwein and Ditor. As a by-product we give a uni…ed proof of the measure and category cases of Unif...

We construct a smooth symmetric compactification of the space of all labeled tetrahedra in P.

In this note, we provide a smooth variational principle on Wasserstein space by constructing gauge-type function using the sliced distance. This is crucial tool for optimization problems and in viscosity theory of PDEs space.