Conventional Barycentric Coordinates are defined relative to vertices defining the boundary of a polytope. Here, we develop barycentric coordinates relative to a cloud of vertices sampling not only the boundary, but also the interior of a region in R d $ \mathbb R ^d $ https://s3-euw1-ap-pe-df-pch-content-public-u.s3.eu-west-1.amazonaws.com/9781315153452/15c305c2-37ac-42a2-a986-bfbc1a242481/content/inline-math13_1.tif"/> , with the objective of using these barycentric coordinates as basis functions to approximate partial differential equations or parametrize surfaces. We show that entropy maximization provides a rational way to define smooth barycentric coordinates, but for the resulting basis functions to be localized, and hence lead to sparse matrices in computational mechanics, entropy maximization needs to be biased by a suitable notion of locality. The basis functions that result from this approach are smooth, reproduce polynomials, are localized around their corresponding vertex, and their definition does not rely on an underlying mesh or grid but rather on less structured neighborhood relations between vertices. Thus, they can be viewed as meshfree basis functions [159,245,380]. 230The theory and practical evaluation behind these basis functions is reviewed next, and selected applications in computational mechanics are presented.