Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: The missing (A, D, r) diagram   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{\'o}
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.

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The categories of this paper are: funct-an
15
The categories of this paper are: nucl-ex
1
The categories of this paper are: cs
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The categories of this paper are: plasm-ph
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The categories of this paper are: q-alg
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The categories of this paper are: math-ph
13
The categories of this paper are: quant-ph
7
The categories of this paper are: hep-th
18
The categories of this paper are: econ
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The categories of this paper are: supr-con
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The categories of this paper are: mtrl-th
6
The categories of this paper are: astro-ph
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The categories of this paper are: stat
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The categories of this paper are: q-fin
10
The categories of this paper are: q-bio
20
The categories of this paper are: acc-phys