Multivariate polynomial splines on generalized oranges (Q6195964)

When piecewise polynomial (spline) spaces are considered in more than one dimension, the identification of the dimension of the linear space spanned by the splines is of the essence (and usually far from trivial). In this paper, polynomial splines are considered on certain types of domains, intriguingly called generalized oranges. An orange is a simplicial complex that has (for a given \(n\) and \(i\in[0,k]\)) \(n\) maximal facs, is defined in \(k\) dimensions and has \textit{exactly one face} of dimension \(k-i\) which contained in each maximal face of the orange. The authors use a projection tool to calculate these dimensions, i.e., they project into simpler, lower dimensional domains called projected oranges, and then compute their dimensions (or, rather, the dimension of the space generated by them).