A note on hot-spots free subregions of convex domains

Abstract The second eigenfunction of the Neumann Laplacian on convex, planar domains is considered. Inspired by the famous hot spots conjecture and a related result of Steinerberger, we show that potential critical points of this eigenfunction (and, in particular, interior “hot spots”) cannot be located “near the center” of the domain. The region in which critical points are excluded is described explicitly.