SIAM Journal on Mathematical Analysis, Volume 53, Issue 2, Page 1670-1710, January 2021.
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 $\mathbb{R}^d$, 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 from both theoretical and numerical results some new conjectures about geometrical inequalities involving the functionals under study in this paper.

中文翻译：

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体积，周长和第一个Dirichlet特征值的Blaschke-Santaló图

SIAM数学分析杂志，第53卷，第2期，第1670-1710页，2021年1月。
我们对Blaschke-Santaló图的研究感兴趣，该图描述了涉及第一个Dirichlet特征值，周长和不同类集的体积的可能不等式。我们对$ \ mathbb {R} ^ d $中的开放集类给出了图的完整描述，基本上表明等距和Faber-Krahn不等式形成了这三个量的不等式的完整系统。我们还针对平面凸域的类别给出了Blaschke-Santaló图的定性结果：我们证明在这种情况下，该图可以描述为包含在两个连续函数和递增函数的图之间的点集。这尤其表明该图是简单连接的，甚至在水平和垂直方向上都是凸形的。我们还证明，填充图边界上部的形状是平滑的（$ C ^ {1,1} $），而下部形状的形状是多边形（除了球）。最后，我们进行一些数值模拟，以便对图的形状有所了解。我们从理论和数值结果中得出了一些关于几何不等式的新猜想，其中涉及到正在研究的功能。