Good neighborhoods for multidimensional van Leer limiting (Q1819089)

Van Leer limiting uses nearby cell-means of a function (integral mean-values -- weighted by a prescribed positive density -- that are taken over each of a collection of nearby computational cells) to restrict the range of values allowed to a linear approximation of the function on a given central cell. These nearby cells -- whose cell-means are actually involved in the limiting -- are called the central cell's neighbors; and the set of these neighbors is called the central cell's neighborhood. The use of certain neighborhoods in multidimensional van Leer [cf. \textit{B. van Leer}, ibid. 23, 276-299 (1977; Zbl 0339.76056)] limiting can force even linear functions to be subject to restriction over the central cell. A simple geometric property characterizes those neighborhoods whose use would not require that any linear functions be limited. (Such a neighborhood is called a good neighborhood for van Leer limiting since its use would not preclude second-order accuracy in the local linear approximation of a smooth function by one that is van Leer limited -- unless the additional, here unspecified, details for obtaining the approximation preclude it by themselves.) The characterization is as follows, where it is presumed that the cells lie in a finite-dimensional vector space: One has chosen a good neighborhood for a given central cell if and only if the convex hull of the centroids of its associated neighbors contains that central cell.