An isotropic Gaussian mixture can have more modes than components

Carreira-Perpinan and Williams (2003) conjectured that a homoscedastic Gaussian mixture of M components in d > 1 dimensions has at most M modes. Prof. J. J. Duistermaat (personal communication, 2003) provided the counterexample of a 3--component mixture in d = 2 where the Gaussians are located at the vertices of an equilateral triangle; for a certain range of variances modes are present near to the vertices and also at the centre of the triangle. In this paper we illustrate the nature of the counterexample and compute the range of variances for which there are more than 3 maxima. We also extend the construction to the regular simplex with M vertices and show that for M>2 there is always a range of variances for which M+1 modes are present.