Number of minima vs. maxima of a periodic function

Question

Consider a continuous periodic function $f$ of $n > 1$ real variables (by this I mean there exists a full rank lattice of symmetries). Does $f$ have as many local maxima as local minima inside one fundamental cell?

Answer 1

A lattice of square-based pyramids provides a simple counter-example: there is one local maximum per cell, but the whole boundary of each cell is a local minimum.

If you would prefer a differentiable counter-example, take $f(x,y)=\sin^2 x\sin^2 y$.