Multivariate polynomial splines on generalized oranges

TL;DR

This paper investigates multivariate spline spaces on a special class of partitions called generalized oranges, providing a method to compute their dimensions using algebraic and Bernstein-Bézier techniques.

Contribution

It introduces a reduction technique for calculating spline dimensions on oranges by relating them to simpler projected oranges, combining algebraic and Bernstein-Bézier methods.

Findings

Dimension formulas for splines on generalized oranges

Reduction of complex spline problems to lower-dimensional cases

Application of algebraic and Bernstein-Bézier tools

Abstract

We consider spaces of multivariate splines defined on a particular type of simplicial partitions that we call (generalized) oranges. Such partitions are composed of a finite number of maximal faces with exactly one shared medial face. We reduce the problem of finding the dimension of splines on oranges to computing dimensions of splines on simpler, lower-dimensional partitions that we call projected oranges. We use both algebraic and Bernstein-B\'ezier tools.