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We examine the optimal mass transport problem in $\mathbb{R}^{n}$ betweendensities having independent compact support by considering the geometry of acontinuous interpolating support boundary in space-time within which the massdensity evolves according to the fluid dynamical framework of Benamou andBrenier. We treat the geometry of this space--time embedding in terms ofpoints, vectors, and sets in$\mathbb{R}^{n+1}\!=\mathbb{R}\times\mathbb{R}^{n}$ and blend the mass densityand velocity as well into a space-time solenoidal vector field ${\bfW}\;|\;{\bf \Omega\subset}\mathbb{R}^{n+1}\!\to\mathbb{R}^{n+1}$ over compactsets ${\bf \Omega}$ . We then formulate a coupled gradient descent approachcontaining separate evolution steps for $\partial{\bf \Omega}$ and ${\bf W}$.

A Variational Surface-Evolution Perspective for Optimal Transport between Densities with Differing Compact Support