论文标题
缺失（A，D，R）图
The missing (A, D, r) diagram
论文作者
论文摘要
在本文中，我们对涉及凸体的面积，直径和inradius的“最佳”通用几何不平等感兴趣。 “最佳”一词应从以下意义上理解：我们解决了在所有给定直径和inradius的凸形集合中，最大程度地降低/最大化凸体的Lebesgue度量。 M. Hernandez-Cifre和G. Salinas在先前的工作中解决了二维情况下的最小化问题。在本文中，我们基于不同的方法对N维情况进行了概括，并在二维情况下完全解决了最大化问题。这使我们能够完全确定平面凸体的所谓的二维Blaschke-Santal {ó}图，相对于欧几里得空间中的三个幅度区域，直径，直径和inradius（表示）（A，D，R）。这样的图用于根据直径和inradius来确定凸集面积的可能值范围。尽管这个凸几何的问题似乎很基本，但直到现在尚未回答。这可能与以下事实有关：该图描述使用了意外的特定凸集，例如一种刻在等边三角形中的平滑非动物。
In this paper we are interested in "optimal" universal geometric inequalities involving the area, diameter and inradius of convex bodies. The term "optimal" is to be understood in the following sense: we tackle the issue of minimizing/maximizing the Lebesgue measure of a convex body among all convex sets of given diameter and inradius. The minimization problem in the two-dimensional case has been solved in a previous work, by M. Hernandez-Cifre and G. Salinas. In this article, we provide a generalization to the n-dimensional case based on a different approach, as well as the complete solving of the maximization problem in the two-dimensional case. This allows us to completely determine the so-called 2-dimensional Blaschke-Santal{ó} diagram for planar convex bodies with respect to the three magnitudes area, diameter and inradius in euclidean spaces, denoted (A, D, r). Such a diagram is used to determine the range of possible values of the area of convex sets depending on their diameter and inradius. Although this question of convex geometry appears quite elementary, it had not been answered until now. This is likely related to the fact that the diagram description uses unexpected particular convex sets, such as a kind of smoothed nonagon inscribed in an equilateral triangle.
