On Relative Geometric Inequalities

Throughout history, many mathematicians have studied and proved numerous geometric inequalities. The first example was the classical isoperimetric inequality in the plane. Later, many variants of this inequality and generalizations to higher dimensions were obtained. Corresponding to them, several relative isoperimetric inequalities appeared. In these inequalities, the area (volume) of a set E was compared with the relative perimeter (the measure of part of the boundary of E, in particular the part of the boundary which is contained in other open set G). The aim of this book is to present a precise study of relative geometric inequalities, in which we compare not only the relative volume and the relative perimeter, but also other relative geometric magnitudes. We shall look for the infimum and the supremum of the considered ratios, and for the sets which attain these bounds (maximizers and minimizers). We shall also obtain relative geometric inequalities for centrally symmetric compact, convex surfaces using the intrinsic distance. Finally, several applications of these inequalities, to other fields of mathematics and to real life problems, are described.