Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely:  quantify conditions that ensure that a signed measure on a Riemannian manifold has a ll defined centre of mass.  then use this result to quantify the extent of a neighbourhood on which the Riemannian barycentric coordinates of a set of $n+1$ points on an $n$-manifold provide a true coordinate chart, i.e., the barycentric coordinates provide a diffeomorphism beten a neighbourhood of a Euclidean simplex, and a neighbourhood containing the points on the manifold.
['Ramsay Dyer', 'Gert Vegter', 'Mathijs Wintraecken']
I am writing a research paper on a idea Barycentric coordinate neighbourhoods in Riemannian manifolds
[{"from": "human", "value": "I am writing a research paper on a idea Barycentric coordinate neighbourhoods in Riemannian manifolds"}, {"from": "gpt", "value": "Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely:  quantify conditions that ensure that a signed measure on a Riemannian manifold has a ll defined centre of mass.  then use this result to quantify the extent of a neighbourhood on which the Riemannian barycentric coordinates of a set of $n+1$ points on an $n$-manifold provide a true coordinate chart, i.e., the barycentric coordinates provide a diffeomorphism beten a neighbourhood of a Euclidean simplex, and a neighbourhood containing the points on the manifold."}]