Q. Can a mixture of a finite number of 2-dimensional normal distributions, with different means and covariances, sum to a constant within some bounded region of the plane?
I believe the answer is No, in any dimension, but I have not proved that, and I hold out some hope that a clever mixture would magically sum to a constant...
I would also be interested if the answer is Yes for any of the commonly used probability distributions: Poisson, gamma, etc.