We are interested in the study of Blaschke-Santaló diagrams describing the possible inequalities involving the first Dirichlet eigenvalue, the perimeter and the volume, for different classes of sets. We give a complete description of the diagram for the class of open sets in R d , basically showing that the isoperimetric and Faber-Krahn inequalities form a complete system of inequalities for these three quantities. We also give some qualitative results for the Blaschke-Santaló diagram for the class of planar convex domains: we prove that in this case the diagram can be described as the set of points contained between the graphs of two continuous and increasing functions. This shows in particular that the diagram is simply connected, and even horizontally and vertically convex. We also prove that the shapes that fill the upper part of the boundary of the diagram are smooth (C 1,1 ), while those on the lower one are polygons (except for the ball). Finally, we perform some numerical simulations in order to have an idea on the shape of the diagram; we deduce both from theoretical and numerical results some new conjectures about geometrical inequalities involving the functionals under study in this paper.

We prove that among all doubly connected domains of  R n  of the form  B 1 \ B 2 , where  B 1  and  B 2  are open balls of fixed radii such that  B 2  ⊂  B 1 , the first non-trivial Steklov eigenvalue achieves its maximal value uniquely when the balls are concentric. Furthermore, we show that the ideas of our proof also apply to a mixed boundary conditions eigenvalue problem found in literature.

In this paper, we prove new sharp bounds for the Cheeger constant of planar convex sets that we use to study the relations between the Cheeger constant and the first eigenvalue of the Laplace operator with Dirichlet boundary conditions. This problem is closely related to the study of the so-called Cheeger inequality for which we provide an improvement in the class of planar convex sets. We then provide an existence theorem that highlights the tight relation between improving the Cheeger inequality and proving the existence of a minimizer of a the functional Jn := λ1/h 2 in any dimension n. We finally, provide some new sharp bounds for the first Dirichlet eigenvalue of planar convex sets and a new sharp upper bound for triangles which is better than the conjecture stated in [30] in the case of thin triangles.

We are interested in finding complete systems of inequalities between the perimeter P , the area | • | and the Cheeger constant h of planar sets. To do so, we study the so called Blaschke-Santaló diagram of the triplet (P, | • |, h) for different classes of domains: simply connected sets, convex sets and convex polygons with at most N sides. We are able to completely determine the diagram in the latter cases except for the class of convex N -gons when N ≥ 5 is odd: therein, we show that the external boundary of the diagram is given by the curves of two continuous and strictly increasing functions, we give an explicit formula of the lower one and provide a numerical method to obtain the upper one. We finally give some applications of the results and methods developed in the present paper.

In this short note, we prove that for every bounded, planar and convex set Ω, one has Funding. This work was partially supported by the project ANR-18-CE40-0013 SHAPO financed by the French Agence Nationale de la Recherche (ANR).