A weighted relative isoperimetric inequality in convex cones is obtained via the Monge-Ampere equation. The method improves several inequalities literature, e.g. constants a theorem of Cabre--Ros--Oton--Serra. Applications are given context generalization log-convex density conjecture due to Brakke and resolved by Chambers: case $\alpha-$homogeneous ($\alpha>0$), concave densities, (mod transla...

We introduce the notion of one-sided mapping cones positive linear maps between matrix algebras. These are convex that invariant under compositions by completely from either left or right side. The duals such can be characterized in terms ampliation maps, which also used to characterize many notions quantum information theory—such as separability, entanglement-breaking Schmidt numbers, well dec...