On the nodal set of the sum of Laplacian eigenfunctions


One guiding principle is the idea that Laplacian eigenfunctions corresponding to larger frequencies should oscillate more. The same principle should then also be true, in some form, for linear combinations of high-frequency eigenfunctions (in one dimension, this is Sturm-Liouville theory). Recent progress on this question is based on the notion of optimal transport and a very simple idea which we formalize: if it’s easy to buy milk, then there are must be many supermarkets (and, conversely, if there are only few supermarkets at least some people have to travel a large distance to buy milk). This turns into a geometric inequality that is interesting in its own right.