On the Cheeger inequality for convex sets
Jeff Cheeger proved in 1970 a famous inequality referring to a lower bound for the smallest eigenvalue of the Laplacian in terms of a constant having geometric significance, called Cheeger constant. The author of the present paper focusses on the Cheeger constant, finding new sharp lower and upper bounds in case of planar convex sets. The proofs are constructive, a detailed diagram being described to generate the upper bound. As a consequence, the Cheeger inequality is improved for planar convex sets as triangle, rhombus and stadium (i.e. the convex hull of two identical balls). The existence of a minimizer in higher dimensions is discussed in more steps: case of sets with increasing diameters and fixed volume, case of of long tubes and finally the proof of the existence result. Finally, some applications are presented consisting in finding new sharp upper bounds for the first Dirichlet eigenvalue of planar convex sets. It is proved that they are better than the existing ones by means of particular cases.
scientific article; zbMATH DE number 7389232
scientific article