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We are interested in finding sharp bounds for the Cheeger constant h via different geometrical quantities, namely the area |•|, the perimeter P, the inradius r, the circumradius R, the minimal width w and the diameter d. We provide new sharp inequalities between these quantities for planar convex bodies and enounce new conjectures based on numerical simulations. In particular, we completely solve the Blaschke-Santaló diagrams describing all the possible inequalities involving the triplets (P, h, r), (d, h, r) and (R, h, r) and describe some parts of the boundaries of the diagrams of the triplets (w, h, d), (w, h, R), (w, h, P), (w, h, |•|), (R, h, d) and (w, h, r). [ABSTRACT FROM AUTHOR]

The object of the paper is to find complete systems of inequalities relating 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 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 boundary of the diagram is given by the graphs of two continuous and strictly increasing functions. An explicit formula for the lower one and a numerical method to obtain the upper one is provided. At last, some applications of the results are presented. [ABSTRACT FROM AUTHOR]

In this short note, we prove that for every bounded, planar and convex set Ω, one has [λ1(Ω)T (Ω)] /|Ω| ≤ π² /12 ·( 1+ √π r (Ω)/ √ |Ω| )², where λ1, T, r and |·| are the first Dirichlet eigenvalue, the torsion, the inradius and the volume. The inequality is sharp as the equality asymptotically holds for any family of thin collapsing rectangles. As a byproduct, we obtain the following bound for planar convex sets [λ1(Ω)T (Ω)] /|Ω| ≤ π 2 /12 [1+ (2 √2(6+π²)−π 2) / 4+π² ]² ≈ 0.996613... which improves Polyá’s inequality (λ1(Ω)T (Ω) ) / |Ω| < 1 and is slightly better than the one provided in [3]. The novel ingredient of the proof is the sharp inequality λ1(Ω) ≤ π 2 /4 ·( (1/ r (Ω) ) + √(π /|Ω|) )², recently proved in [8]. [ABSTRACT FROM AUTHOR]

We are interested in the study of Blaschke-Santal\o diagrams describing the possible inequalities involving the frst Dirichlet eigenvalue, the perimeter, and the volume for diferent classes of sets. We give a complete description of the diagram for the class of open sets in Rd, 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-Santalo 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 fll the upper part of the boundary of the diagram are smooth (C1,1), 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 from both theoretical and numerical results some new conjectures about geometrical inequalities involving the functionals under study in this paper. [ABSTRACT FROM AUTHOR]