An enhanced uncertainty principle for the Vaserstein distance

We improve some recent results of Sagiv and Steinerberger that quantify the following uncertainty principle: for a function (Formula presented.) with mean zero, then either the size of the zero set of the function or the cost of transporting the mass of the positive part of (Formula presented.) to its negative part must be big. We also provide a sharp upper estimate of the transport cost of the positive part of an eigenfunction of the Laplacian. This proves a conjecture of Steinerberger and provides a lower bound of the size of a nodal set of the eigenfunction.