On Delaunay oriented matroids for convex distance functions (Q1923774)

In the \(n\)-dimensional Euclidean space a finite point set can be partitioned into a polyhedral complex by such hyperspheres which are determined by \(d+1\) points of the set and don't contain any other points from the set in its interiors. This complex can be regarded as an oriented matroid of affine dependencies of the point set of the \(d+1\)-dimensional space obtained lifting the original set into the paraboloid of equation \(x_{d+1}= \sum^d_{i=1} x^2_i\). The author proves the following statements: 1. The above matroid can be defined in the case where hyperspheres are substituted by the collections of \(K\)-circles of a smooth, strictly convex distance function \(D_K\) in the plane. 2. The \(K\)-Delaunay oriented matroids so obtained may be nonrealizable. 3. These distance functions do not have, in general, a lifting property.