Isotropic Positive Definite Functions on Spheres: Even-Dimensional Gneiting's Problem

We study when a compactly supported isotropic positive definite function on a Euclidean space remains positive definite after Euclidean distance is replaced by spherical geodesic distance. The corresponding assertion is known in odd dimensions. We prove that it fails in every even dimension, no matter how small a positive upper bound is imposed on the support. An explicit counterexample is also given in dimension two.