Topology of Singularities of Optimal Semicouplings

This article is a summary of some results from the author’s thesis [Mar]. We study the topology of singularities of c-optimal semicouplings in unequal dimension. Our main results 1, 2 describe homotopy-reductions from a source space (X,σ) onto the singularities Zj , j ≥ 0 of c-optimal semicouplings whenever (Y, τ) is a Riemannian target space and c : X ×Y → R∪{+∞} is a cost satisfying some general assumptions (A0)–(A5). We construct continuous strong deformation retracts X ; Zj whenever a condition called Uniform Halfspace (UHS) condition is satisfied along appropriate subsets.