Blaschke--Santaló Diagram for Volume, Perimeter, and First Dirichlet Eigenvalue

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