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We show, by an explicit construction, that a mixture of univariate Gaussian densities with variance $1$ and means in $[-A,A]$ can have $\varOmega (A^2)$ modes. This disproves a recent conjecture of Dytso et al. (2020, IEEE Trans. Inf. Theory, 66, 2006–2022) who showed that such a mixture can have at most $O(A^{2})$ modes and surmised that the upper bound could be improved to $O(A)$. Our result holds even if an additional variance constraint is imposed on the mixing distribution. Extending the result to higher dimensions, we exhibit a mixture of Gaussians in ${\mathbb{R}}^{d}$, with identity covariances and means inside ${[-A,A]}^{d}$, that has $\varOmega (A^{2d})$ modes.

How many modes can a mixture of Gaussians with uniformly bounded means have?